@IdontWanna
There are various options the OP could do.
Underlining them all is to get over to this violent person just how serious the situation is and how it will not be rewarded by kindness.
I would inform the police. Would they do anything? I hope so but, whether or not, I would march her down to the station first thing in the morning to frighten the shit out out of her. if it doesn't frighten her, then she really is on the path to God knows what.
I might inform the school. Not only would I tell I was doing this, I would have her there while she I did it.
I would make her apologise to her brother and, whether she meant it or not, I would reward her brother at her expense...fining her if you like. Something like he has her pocket money for a month or cleans his room every day.
I would no nothing for her beyond feed her for a full week. No Tv, no nothing that would bring her a moment's pleasure for a full week.
She would know she is in disgrace and I would be making no excuses for her. She has placed herself outside the family and she would feel that.
I would let her know that if she tries to crack a stranger's head open, she would be placed outside society by going to gaol.
Would I tell her friends and wider family? Maybe?
I would do anything that would hopefully make her ashamed and uncomfortable both for the sake of my own family and wider society.
If she couldn't be shamed then I would know exactly what type of person she is. The type of person who kicks a tramp's head in because she can and I would watch this dangerous person like a hawk.
She beat her mother up and she is only 11. Do you think violent criminals don't start their career until they have turned 18? Violence is violence and this person threw her mother to the floor and, not content with that, punched her head three times. It wasn't a fucking accident or moment of madness. She carried on!
What I wouldn't do is crawl around after her. The OP has done this and what was her reward...an unapologetic face and attitude. So that worked, didn't it!