All these posts and replies, and what we still aren't clear about are the specific terms of OP's landlord-tenant relationship. Who owns the property? What are the landlord-tenant rules in your state?
Based on OP's changing testimony, we now gather that the addled woman is more mentally ill than demented. In most states, conservatorship for mental health is a different game than for other types of disability, including dementia. She would need a documented history of mental health issues. She would require a psych eval. And yes they are more difficult to obtain.
By cleaning her house, you're postponing the reckoning, OP. If you want people to take you seriously, you're going to have to stop helping her. Don't do anything except take pictures of the gathering piles of trash. Record all your conversations with her and save them, if you live in a one-party recording state.
If as you claim OP, people are telling you to get a POA, ask these people to actively assist you in obtaining one. Rather than calling the Area Agency on Aging, call NAMI, the National Alliance on Mental Illness, and get their take on things.
Speak to your father, and tell him that due to the woman's ongoing troubles, that you cannot keep her in the house, for whatever spurious reasons you have. You've come up with many such reasons in this thread for instance. If he owns the house, then you are probably out of luck.
Nobody can legally fault you for neglect, because she is legally a competent independent adult, and she has not been assigned to your care. The attempts you have made are sufficient to show you are concerned; but be clear there are no legal obligations that can be enforced here.
You can also speak privately to the crazy woman, and explain that you may have to evict her if she doesn't comply, due to her unspecified problems. With what she needs to comply, I'm not certain. Then you explain that you will need a POA so you can help her comply with the unspecified compliance measures. I'm sure she'll understand better than any of us on here.
You most certainly can evict her if that is an option available to you. At which point, as the police and welfare calls mount, you might be able to start forcing the issue with APS and other agencies, explaining that you tried to help, but nobody would assist you, and you had to let her go because you need legal (POA, conservatorship) and other supports to make the situation work, and can you please help me with this, because you can't have a mentally ill woman paying no rent and trashing your house? You might get a different response after the woman becomes homeless. Might.