Good morning, Todd!
Well you are in a right mess, aren't you? It's too late for Power of Attorney given the diagnosis of dementia. Guardianship is the path to go, i'm pretty sure. The nursing home she's in should be able to guide you somewhat: is there a social worker there? Or perhaps the administrator is familiar with this process? Most long term care facilities very much WANT to have a family member as next of kin with guardianship.
I would try the following resources for help with this process: look up www.eldercare.acl.gov and put in her zip code (of her apartment), and then call the agency that pops up. Ask for information on obtaining guardianship for your mother who has dementia. I actually called up the Legal Aid Society of New York and when i got through the maze of press this for that and that for this, i got a live human who said they do not help with guardianship cases. So my guess is the area agency on aging through Eldercare.acl.gov is your best first step.
Start there and then write more? Your mother is blessed to have such dedicated children, and i wish it wasn't so hard.
My other thought at the moment is to call your congressional representative's constituent services person. Or other local government official. This is a matter of public benefits and rights, and there has to be a pathway forward.
Write more?
Jane