I surely hope you mean by "here" that that isn't in your own home. My mom is 3 miles away which is enough to gauge when I can go see her etc. Getting her social help was huge for me, as I'm disabled myself and the rest of the family of course felt since I lived so close it was my job to "take care" of mom. Not.

Does your sister know of such abuse? Your mom can choose to live wherever she wants to, it is a free country. Unless you can get her evaluated and into the social system up where she is, so she's less likely to leave it, she just might move. Is she confused enough to keep correcting her that she's going to your sister's and not to your city??? If both you and your sister work from the same angle that will work for you. Your mom will need help moving, yes? So you and your sister just help her move to your sister's, if she continues with this idea. It does sound like she's confused the efforts.

Breathe and breathe some more. This isn't the end of the world. You still have your own life. You will not have to drop anything to take care of or tolerate her, just as if she were miles away.