I don’t know if she is mentally ill, but she is certainly distressed and distressing to be around. You have not been invited by her to be helpful to her so chances are that any help you offer will be badly received. You’ve described a woman who doesn’t take responsibility for herself and who seems to think that whatever challenges she has to face are someone else’s fault. To her, you will be no different.
I think all you can do in the situation is support your girlfriend in her efforts to get on with her own life. She cannot reason with a mother who is being irrational. Trying to talk her mother out of her opinions is a futile effort.
Your girlfriend is not obligated to put up with abusive language or behavior from her mom. Her biggest challenge is learning how not to fall for the bait by fighting back. She’ll get further and will feel better if she simply says to her mother that she will leave the scene if her mother tries to engage her in a fight or calls her names. Then she has to do it. Leave.
It’s very difficult to give up on a parent. But sometimes it is only by giving up that things start to get better. Her mother has to realize for herself what she has lost by alienating her whole family. Only then will there be an opportunity for change.
I wish you well.