Assume a strong position of authority. It's your apartment, and you have the right to decide who stays there. Flat-out tell her to leave, making it clear that you mean business. If she continues to refuse, threaten to get law enforcement involved.
I think this is a better solution than trying to break them up. Rather than deliberately ruining your brother's relationship, you're just dealing with the problem in your immediate life, so he can continue to see her if he chooses to. If she's as shitty as you're describing, eventually they will break up on their own and he will see your superior wisdom.