I agree it is not a good idea for you to be bringing your mom to your house, because you are a single guy, and think long-term care is the best option, but DAMN! this statement annoyed me! I am a nurse, and therefore my ex never felt the need to involve himself in the care for my daughter, who has a chronic medical issue. Taking care of your own family is NOT the sole responsibility of the family nurse, and quite frankly, day-to-day care of dementia patients is NOT what a nurse does. That is what unlicensed caregivers do.
Your sister should be compensated for the impact your mom's care has taken on her life thus far, and your mom should go for long-term care, unless there is someone who is willing and able to step up. Frankly, I wouldn't do it for a violent person. I've been given a black eye by a 93 year old man with dementia. They CAN hurt you.