This kind of situation is terribly difficult. However concerned you are, you can’t “make” your mother take care of herself. Although I can’t offer a diagnosis on the basis of a letter, it is probable that you are correct, that she is depressed.
What saddens me most is that she is letting her former husband “win”. She may think she is making him feel bad by behaving in this way, but she is incorrect. From what you shared in your letter, he probably doesn’t care.
There is a famous quote by a British poet named George Herbert. He said “Living well is the best revenge.” The way to recovery for your mom is to reclaim her self-esteem and to show her ex that she is fine. She can do that by taking care of herself, by finding something to do that gives her life meaning, and by generally moving forward in her life.
Even if your mother refuses to see a counselor, I encourage you to do so. You are a sensitive and loving son. You would probably benefit from getting some practical advice from someone who can hear more of the details as well as some on-going support in this difficult time.
I wish you well.