your mom needs to leave. she hit your child. she should not be in your home. end full stop.
I know you will probably give me an excuse about why she should stay. but she is no longer welcome at your home and when she hit your child she crossed a line that was never to be crossed. she is a grown up. she needs to find somewhere to live. she has 800 bucks. she can find a roommate and get an apartment. get food stamps. you will be fed on food stamps. the rest of it is not your concern.
she needs to leave like yesterday. your daughter needs to be protected. you fought to keep her in your house and keep her safe and clearly she is not safe from your mother.
take a deep breath and tell her she needs to go. pack her stuff and show her the door. you can do it.
just thinking out loud here, if you are concerned for your safety, then you should call a friend. give her 24 hours notice. I might even give her less if I were you. find a homeless shelter? whatever. she needs to go.