Abusive Child (14) In Michigan, What Now?

My sister (14F) violently abuses my mother and sister (15F), destroys the house, and screams/cusses and demands for what she wants daily. My sister has taken many medications and has been in therapy basically her whole life (she’s behaved like this since she was a toddler), her therapists deem her actions as behavioral, not mental. My mom has tried calling the police, she’s been with her to the ER countless times, and no hospital can admit her since she punched my mom in the face while on camera in the hospital, and is deemed violent. My sister goes into court for three domestic abuse charges against my mom, sister, and mom’s bf in two weeks, where the maximum they can give her is probation. My mom is now looking into giving her up to the state. What other options can she do that is long term inpatient? They live in Michigan and this seems to be uncommon in the small town she is in, if anyone knows any programs or further steps she can take within the state, it would be gladly appreciated.