I moved my mother into my house to help take care of her, but she hoards things and makes cluttered messes everywhere she goes so much so that you can't even clean under things anymore. It's just piles or junk everywhere. I basically just live in my bedroom now and only come out to eat and go to work. My house is no longer mine and I will have to make major monetary investments to sell it one day. She's completely ruined the floors and spare bathroom. Her dying cat urinates anywhere it wants and even bringing up the suggestion of putting it down causes my mother to go into crying tantrums. She threatens to move out when I bring these things up and tries to play the guilt trip that she needs help and nothing is her fault. I just can't take it anymore. I want to move and never come back, but it's MY house. My mother pays for nothing.
Take her for a tour and free lunch at a nearby assisted living facility. She might even want to go.
Time to stand up to Mom and tell her it is YOUR house, YOUR rules. The clutter has to go, or she will need to move. The deadline will be March 31st. If Mom does nothing, then she will need to leave. I know she will throw a fit, and if she does, turn around and walk away.
Check to see if there are "senior apartments" where the rent is based on one's income. She might be able to budget so she can live on her own with low rent, and enough left over for food, etc. Plus she will be around people from her own generation.
As for the cat, what is the health issue of the cat? Peeing all over the house, could mean the poor thing could have an urinary tract infection which is treatable. Some cats will squirt on the wall to get the owner's attention that they need some medical help. Mom or you need to get the cat to the Vet.
If it's not a workable situation with her living with you, then, I'd assess if she's able to live alone and take care of her needs and the cat's. Call your local counsel on aging or social services to get info and to see what is available for those with income restrictions. If there is no real viable way for her to set this up on her own, find apt., pay rent, utilities, etc. ,then, I'd try to help her get it arranged. Continuing the struggle or war on words won't help. Just try to get it worked out, so both of you can find some peace.