If your mom's Alzheimer's prevents her from making healthcare decisions on her own behalf she isn't competent to sign a POA. I agree with Felidae, you probably need an attorney and you probably need to seek guardianship at this point.
Have her interviewed by an elder specialist for competency. Sometimes the seniors are not too far gone. Having a good POA, will, trust, etc documents is easily understood, even if the person has some impairment. An elder specialist can help clarify the issue.