Since she is not doing anything to help her. My sister, while my mother was living with me, printed a DPOA form off the internet, had my mom sign it and had it notorized. At that point my mother was not capable of making those decisions as she would have signed anything for anybody. About a year later I had to move my mother into a nursing home since I had to have a hysterectomy and my sister would not help me care for her while I recovered. My sister eventually moved my mother to a NH in Indiana where she has no family and my sister is only there 5 months out of the year so my mother is alone there. I want to move my mother back to Michigan to be by me and family. They have drugged her up so much she cannot walk or talk and I need to get the power to move her back home to a NH here where I can help her. What do I need to do to prove the DPOA was illegal since my mother had Alzheimer when she signed it. Is the DPOA signed in Michigan even legal in Indiana?
So. Find your placement, pay your deposit to reserve the room. Contact her current NH. Get everyone's agreement in principle to the move. Get the logistics worked out to the last detail. Then present the proposal to your sister, cross your fingers, and hope she'll agree to it. Any reason why she shouldn't, really? And since neither of you, apparently, has or can obtain POA you can rely on your status as next of kin - hence, much easier if the two of you are agreed.
Having said that, I think you might find your sister's POA is accepted as valid. The time to object was when it was being done. Seeing as it was notarised, and nobody demurred, you're going to have a job proving that your mother lacked capacity back then.
The other thing you don't mention is how long your mother has been in her current NH. Are you allowing for how hard the impact and disruption of a move could be on her?
Your mother has been sort of painted into a corner, and I do sympathise, and this situation must be both depressing and frustrating for you. The question is whether there is anything that can realistically be done about it at this stage. The move may not be impossible, but it's going to be tricky, expensive, of dubious benefit to your mother, and fraught with potential for further fallings-out.
Is money an issue?
However, first I'd follow the excellent advice that Churchmouse gave. It is always easier if you can get siblings to work together instead of fighting them AND this dreaded disease- dementia. If it were me,'I would also take the additional step of at least seeing an attorney regarding guardianship. Then if your sister doesn't agree to letting your mother return to being near you and other family - you can give your sister your attorneys business card and let her know you are serious about pursuing moving your mother home.
So if you can't negotiate with your sister, then guardianship would be the way to go, as Rainmom explains. But do think it right through first, and be sure you'd want to take it on, and that it would have a good outcome for your mother.
I have to say your sister sounds suspiciously like the sort of person who thinks that POA gives you Power Over a person, rather than power to act for them. How long has she been exercising it for your mother?