What happens to our lease agreement when my mother, whom I live with, is placed in a care facility?

Cap, personally, I would get the damage fixed now. Then you aren't turning over a damaged property and mom can pay the handyman now.

As a rental property owner, I don't care who tore my property up, whomever is on the lease is going to pay to have it repaired, that is industry standard.

I highly recommend white vinegar for killing the enzymes in the urine that will show up on a black light test.

Once you have the work completed, you can have mom buy a prepaid credit card that you can hold onto until you both have relocated, in the event you need the handyman again.

Please check with a local lawyer to best advice. Usually, both parties are responsible for maintenance of the property as well as damage. Check your lease for what happens if you break the lease early.

It is too late now for POA so it is time to seek guardianship now. That will go uncontested and you get it fairly easily. Your use of the word "Mum" makes me wonder if you are in the UK? In the USA if your Mum goes to hospital the social workers can easily get you a grant, help you get one of temporary guardianship, which can be made permanent easily with an elder law attorney. You will then be responsible for all Mom's bills and finances, and will pay them out of your own accounts for your portion. You will need to keep meticulous records of all this. As to your Mom's name on the lease, you can explain she has descended into dementia and can no longer be responsible; you may have to pay out of her funds that portion on the remainder of the lease, but that will be over soon enough. And yes, damages will be paid that she did out of her finances.
Best of luck. See an elder law attorney for your options here.

In this country (US) at least, both parties are responsible for damages to an apartment, and it usually comes out the the deposit (bond) you put down when you rented it. Same goes for the rent -- both parties sign the lease, so both are responsible. The landlord doesn't care which pays the rent, but if no one does, they have two people to go after to get their money

Probably the only way to get your mother off an existing lease is to provide proof she's no longer competent and has to move to a facility. That might break the entire lease (check the lease terms) and you'd have to re-qualify on your own, or more likely, you're left with paying all the rent until you move.

How much time is left on the lease? Can you sublet her half of the apartment for the remainder of the lease?

Since a lease is a legal document between parties, and to the best of my recollection is typically witnessed and notarized, changes can only be made by the original parties to the lease.   I doubt you have any authority at all to remove your name w/o either amending or entering into a new lease. 

You need to amend the lease to accomplish the changes you want, but you also need to address the issue of it being considered a changed arrangement and/or new lease if you remove your name.

There might be the possibility though of creating and executing an exculpatory amendment to remove your name, and your liability.   This should be done by a real estate or landlord/tenant attorney.  It's not a job for someone w/o legal experience.

EDIT: I did not realize you are in the UK. My suggestions obviously were based on the U.S. I'm going to leave them here anyway as some of them might translate over into UK law.
1. Read the lease. Are you a joint tenant or co-tenant? Does the lease say anything about going from the current lease period to a month-to-month arrangement?
2. Check with an elder law attorney about the laws in your state. Some states allow tenants or co-tenants to terminate a lease because of disability.
3. Check with community eldercare organizations. This problem has probably come up before.
4. There are many stages of dementia. One can have what we call dementia and still have capacity to make legal decisions. I urge you as soon as possible to see an elder law attorney, who can help determine her capacity. She may still be able to name you as her financial/medical durable POA agent.
5. Has she seen her doctor or a doctor who deals in dementia? If not, do this right away.
6. I don't know for sure, but I imagine a person without legal capacity can't legally pay from her accounts. I could very well be wrong but her incapacity may be the same as if she had moved out. Look at the lease to see what your responsibilities are as a co-tenant when the other tenant has moved out.
7. Do you have a relationship with the landlord? Can you talk to them about this? If there's a property manager, try to talk to them too.
8. Do as much repair up as you think is worth it.
9. If she cannot assign powers of attorney, you will probably have to petition the court to become her guardian/conservator. This takes time and does cost money but I believe you can be reimbursed from her accounts.
10. I don't know what state you're in. State laws and regs vary. You must get legal counsel where your mother lives.

NOTE: A person can sign durable medical and financial powers of attorney at any time after they are an adult as determined by your state. One does not have to be assessed. In fact, have a trust and estate attorney draft a complete estate plan once you're of age as long as you are competent. Then have your plan documents updated on a regular basis, perhaps every five years.

These are suggestions and not legal advice.

Connect with a local social worker and discuss your situation with them. They may be able to give you ways to connect to a pro bono attorney who specializes in elder law. If your mother's dementia is not too advanced, and she is still able to sign legal documents, it would be best for you to help her get all of her paperwork in order. It will make things easier in the long run. Usually you'd want to have her set up POA for medical and financial matters, a living will with her advance medical directives, and a will if she has assets. At least try to be on file with Social Security and Medicare so that you can speak on her behalf. You can do this with a phone call with her sitting next to you to agree to some verbal statements. If she has a credit card in her name, get a second card on her account with your name on it so that you can purchase things for her. If your mother will agree and can sign the papers at the bank, it will make things much easier if she will make you a joint owner on her bank accounts and any other financial accounts she might have. Then you will be able to sign checks, etc. Also, the living will is important. Do you know if she wants "do not recuscitate" and to avoid heroic measures (such as ventilators, feeding tubes) to keep her alive? My mother's assisted living/memory care facility wanted to have a copy of her advance medical directives to post in her room so that EMS could see it, if they had to come to assist her. If it's too late for her to sign legal documents, put her medical directives in writing, and get her to sign it if she can. You can do the same with a will. It sounds like could be dangerous leaving her alone. Try not to let her use the stove, and try to check that the faucets are turned off when she uses water. Talk to the social worker about having caregivers come in for visits as often as possible so that you can get breaks and have someone watching over her when you are not there or have to work. This is especially important if she is a wanderer (would leave the house and get lost). My mother mis-used the microwave and broke it, and some of the other tenants at her facility forgot things on the stove and set off fire alarms. If dementia is advanced, it's also best not to let her use sharp objects or machinery (knives, scissors) - her judgment may not be good about them. For the incontinence, start her using adult panties for incontinence and get the disposable pads to put under her on her chair and bed. They are usually found in the adult care sections of stores. Professional caregivers can also advise on this. Regarding the lease, you will be responsible for damages as co-signer. I think it's best to leave her on the lease until she moves, and if you'll be moving then, it won't be worth changing the lease (especially if you want payments for damages to come from her account). An elder attorney could help you with this question. Usually landlords expect some minor wear and tear, such as needing to paint when a tenant leaves. Definitely try to keep urine (and worse) off the carpets, etc. Try to avoid any urine smells. All the best to you both.

If your mother is also on the lease she is responsible to pay for her half of the cost until the lease runs out whether she goes into a care facility or not.
She can be taken off the lease but will have to pay a cash penalty for breaking a lease contract. People often don't know that when they're placing someone in facility care, they don't have to just hand over every cent that person has. If they person going into care has unpaid bills in the outside world, they have to be paid.
NH, MC, AL, whatever kind of care facility will always try to convince a family that their bill must be paid in full first before they can even consider paying any others for their loved one. That's totally not true.
Take your mother to a lawyer and have Durable POA documents done. Any lawyer can make these up (paid for by mom). You don't have to pay more by having them done by a lawyer with a specialty in elder law.
Once that's done and you've placed your mother, start paying her bills. Like her remaing share of the lease. Or her share of repair or cleaning for the apartment when you move. Her credit cards, etc... Most importantly, make sure you do a pre-paid funeral contract paid for by her funds.
After you've gotten POA you can start billing for caregiving services that you provide and also for any services you carry out as POA. Like making phone calls for her, handling her medical/financial affairs, taking time out to tour facilities for her to go to. None of this has to be done for free. These are outstanding bills to be paid for out of her funds.
When you go to have the POA done, ask the lawyer about this.

I was worried to death about asking my 93 yr old mum about resuscitation. I finally broached it saying doc wanted to fill some forms and that was one of the questions , all the others were a matter of history but, I said this one you need to answer
I explained they told me resuscitation is very tough and often means intense bruising an broken ribs . She just told me she wouldn't like that at all just please let her go and then we moved on to other things on the form I was so relieved, such a horrible thing to ask your mum but so very necessary.

Sometimes (note “sometimes”) you can break a lease if you present the situation with management and ask for an exception. Everyone knows that dementia is devastating to all who are related to the person, so you might get a break then. I’d approach it gingerly with a distressing face. That might get you a pass.