I wish they had posted more info. If the rule went into effect after she was there with her dog, I don't think she should have to get rid of it and there may be a 'grandfather' clause that allows her to keep that dog but not replace it if something happens to it.
If she moved in knowing no pets were allowed, why not find a place that does allows them? Isn't it easier to find places that allow small dogs rather than big ones?
My great aunt was in a nursing home in the Alzheimer's Unit and they had a dog that lived there, too, to keep the residents company. We always brought our dogs to visit my aunt - she didn't always remember us, but she always remembered the dogs!