Your mom can't appoint POA if she isn't competent. There are others here who know way more about this than I do but I think you're going to have to gain guardianship over your mom via an elder law attorney. Hopefully those who have been through that will stop in and share their experiences.
What is the sad part is that most of us don't know that when our parents get older, that our parent needs a Power of Attorney for medical and for finances, plus a current Will, and a Living Will which lets everyone know what that person wishes to do if in an emergency, or Health Directive.
None of the above is ever taught to us in high school or college, thus it's like going into a "new job" without any training.
All we can do is learn from our parent's mistakes and make sure that WE have all our own legal documents signed so that our own children or relatives know what to do.
What is also sad is that if our parents wait to sign such documents: "I'm just not ready to do that yet," then it may soon be "too late" as their dementia descends into paranoia....and they they definitely don't want to give power to their kids "who may be up to something" like moving them into a care facility!
A good attorney will go to the Nursing Home and talk to mom. If mom is intact enough to understand that she is putting you in charge of finances, the POA will be executed. The Nursing Home would also need a Health Care Proxy for you to talk to them about medical stuff. Do them both at the same time.
None of the above is ever taught to us in high school or college, thus it's like going into a "new job" without any training.
All we can do is learn from our parent's mistakes and make sure that WE have all our own legal documents signed so that our own children or relatives know what to do.