You did mistake by apologizing where you were not wrong. When you accept her bullies like this, it is at the cost of self respect. Now she knows that she is your weakness and you are not able to live without her, naturally she will show her attitude. You should make your stand absolutely clear on your parent’s issue and should not tolerate her misbehaviour. I’m not telling you to fight with her but at the same time you have to defend your self-respect at any cost. While changing your attitude like this, she may throw tantrums against it. You have to be firm on your stand.
If her mother is suffering from some mental illness, then there is a possibility that she may have inherited bad temper or personality traits.