I don't know if I can take care of my mother at home anymore. How can I get her into a nursing home?

My mother is currently in rehab following a hospitalization for pneumonia. My problem is that I don't know if we can keep on taking care of her at home. Dad is 83 years old and has been her caregiver, taking her to the bathroom, getting her up and putting her to bed, getting up in the night with her etc. She is unable to take care of herself and will walk with a walker only very little. She refuses to exercise unless we make her and take her through the exercises herself. She will only walk and do "laps" around the house if she is made. As a result, she is grossly overweight and getting weaker. I live with them and do the housework, laundry cooking, etc. But it's taking a toll on me. She has a temper and has called me bad names. She'll have massive diarrhea or bm in the middle of the night and I'm up helping clean her up and doing laundry. She doesn't want to throw anything away, so it's collecting up. ARRGGGHHH!!! Do I alert the rehab center of the problems? Would they just keep her?