Should I forgive this kind of person after she apologized?

You should forgive her. You're not perfect either. We all have our flaws. Tell her you forgive her but until she treats you better you two can't be friends. If you don't forgive her then that's just something she could put on her list of why she doesn't like you. Show her how a good friend acts. Maybe she'll try to copy your behavior. It couldn't hurt, right?