What if mom won't sign papers to sell the house for nursing home memory care?

I agree with others you recommend you read your PoA document to see when your authority is activated and what your powers or representation are. If there's any question at all about whether or not you should move ahead with the sale, please consult the elder law attorney in advance.

See if the facility Dr will declare her incompetent to make informed decisions. Read your POA, if immediate you can sign papers for Mom. Mine was written I could buy and sell. See if it says needs a Drs delaration of incompetence.

As POA you take the place of Mom. Just make sure POA follows ur name.

POA has NO POWER over a Trust. As with my own case, my brother having both accounts not in trust, and property and home IN Trust I had to be made both Trustee of Trust and POA. Two sets of documents I had to use interchangeably.
If your mother is no longer competent to make you Trustee of Trust then read the Trust to find out when you can be made Trustee if you are now listed as successor Trustee.
This really should be done with you and an elder law or Trust and Estate attorney together, because you may need to be guardian in order to accomplish sale and funds placed for payment of Mom's bills.
Your mother is able to keep a home and a car and whatever sum your state allows.

And AGAIN, bottom line here is that you cannot sell or even list a home until you are TRUSTEE. You cannot do this with a POA.

ALL of this and so much more and more accurate info is in the offices of your attorney. Your POA pays for expert advice and you NEED and must have good professional advice, not the opinions of a bunch of us on a Forum. As Igloo often says, most of this is no longer DIY stuff.

Do understand, if your Mom is competent under the law all of this is still her own choice. That makes it quite complicated as there really are NOT a lot of choices here.

You should talk to the lawyer about getting conservatorship over your mother. This way you won't need her signature for anything. You actually don't need it now if you have medical documentation from a doctor stating that she is incompetent. She already has a lawyer so one will not have to be appointed to her to represent her for the conservatorship. You will need medical documentation from her doctor that she cannot make decisions for herself anymore then the lawyer will do the rest. The lawyer's say at a conservatorship hearing is usually what decides it.

Your mother will likely 'showtime' for the probate judge at the conservatorship hearing, but they see this every day. If you have documentation that she is mentally incompetent, then you will be awarded conservatorship. Then you sell the property.

Documented Alzheimer’s makes her unable to sign legal documents despite any “good days” She named you POA for exactly a time as this, she’s unable to make reliable decisions in her best interests anymore. She trusted you for this job, and it’s always a good idea to have an elder care attorney walk you through the next steps