I'm going to dump a mentally ill girl
Why would you dump her?
Her actions and attitude gives me a headache and she can't even take care of herself.
You have the right to be with who ever you want, so you shouldn't feel bad if you want to break up.  If you talk to your partner in a respectful way, she should understand your position and you two can have a mutual break up if it's not working out.  Talk to her before you dump her to see if you can resolve the issue first so that no one gets emotionally hurt.
Fine, I'll break up with her but she'll get the blame for it.
She shouldn't be blamed for your breakup. Please understand that people can't help their mental illnesses.