It is important to act quickly and address these questions. What is the criteria for legal capacity to sign a Durable Power of Attorney in your state? Does your mother still have the capacity all or part of the time?
An Elder Law attorney can explain the criteria for legal capacity. Your mother's doctor will have to decide if your mother meets the criteria. If she does, your attorney should move quickly to finalize the power of attorney and other documents you will need to care for your mother as her dementia progresses.
It is possible that a person can be unable to manage their financial affairs, while being able to make a sound decision as to who should manage their finances for them. But dementia is unpredictable, and I cannot over stress the need to take action immediately.
I can tell if a person understands what he/she is about to sign. Early stage dementia sounds like she is competent to sign a power of attorney. If I were licensed to practice law in your state, I would talk to you over the telephone so I could prepare the document in advance. I would go to the nursing home and meet you and your mother. I would chit chat with both of you for 5 to 10 minutes. When your mother felt comfortable, I would ask you to leave the room and go get a cup of coffee, not hover near the door. When I was alone with your mother, I would make certain she understood that she trusted the agent (you?) with her money and other assets. If I was certain she trusted the agent(s), I would have her sign the power of attorney. If not, I would find out who she really wanted. I have encountered that issue from time to time.
If I was unsure whether or not she was competent, I would ask the social worker to join us. Sometimes people have communication issues, but have enough competency to sign a power of attorney. Of course, if she didn't understand what she was about to sign, I'd have to tell you it was too late. In that situation, she might need a guardian. This is why I tell people to get the health care proxy (whatever it is called in your state) and the durable power of attorney asap.
I charge $150 for the document and my hourly rate for my travel to the nursing home. I recommend that you call a few different attorneys and find out what each attorney charges for the nursing home visit.