Wendy, hang i there! Until your sister gets ill she may not understand your Mom's weakness but taking her in is still an act of love for you and your mom, to give you both a chance to heal. Just keep telling your sister that your mom has many miles on those bones and needs more rest than folks think. If she goes with her to physical therapy, she can talk to the PTs about it if she thinks your mom is not getting her strength back fast enough. She needs peace and quiet and hopefully, your sister's house will give her that. Your sister needs to feel useful so even if she gets snippy at you, be sure to tell her to go gentle on your mom.
I do hope you and your mom get better soon. Eat as many fresh fruits and veggies as you can. Lots of water. Pace yourself when you do have energy. Keep loving. And keep posting in here. Not only is it good for you, it is good for us to hear how you're doing.