As you and Glenda and Jane have all said, you can't change your sister or her husband (who really seems to be the one having problems with the situation). The only thing that I see you might be able to do is offer them a respite, and take your mother back into your home for a brief period, or possibly start a rotation with them of so many months in each home. I don't know whether this would be practical or even desirable for you.
We had my mother on a three month rotation for a few years before she became a fall risk and went to assisted living. My DH was the one that couldn't handle having her here all the time. My home was the permanent address. She spent winters with my brother in S. Fla. and summers with my sister in Colorado. We had her in the spring and fall. This spread the burden of her physical care, as well as the financial cost of feeding her, etc. It also spread the joy of having Mom in our homes - which was especially nice for my sister's kids.
Just a thought. Either way, I wish you well.