She is only a vile child because she is allowed to be. If you make the mother deal with it she either will or you have the ability to walk away. The more you react to her the more she will perform, because she gets a reaction.
Ideally her mother should pack her bags and tell her to either unpack them again and behave or take them with her as she leaves. If mother won't do that tell her to pack yours. see what happens!!
I promise you, you don't want to let it get to you as it appears to be. I've been there and all the misery now seems worth it now because of your relationship with the mother but you will look back on it as time wasted as you get older. Give them an ultimatum and stick with it. If you don't you will regret it later.