HOW CAN I GET MY MOTHER TO UNDERSTAND SHE CAN NO LONGER LIVE ALONE?

It depends on the person. If your Mom is very strong-willed, she may not understand this yet. Usually it takes a while and she has to be there to realize that. e.g., my father-in-law didn't want any caregiver or live-in at all - he only wanted to live with us. But he didn't realize he cannot be alone. Now he is ok with it because he knows someone has to take care of him - not that he is clear why. At times he still does not want the caregivers considering money and privacy. But as things get worse, he has no choice.
In your mom's case, perhaps she would want to wait until she really cannot be alone. You need to watch her carefully as she would still want to be alone. Also home care service administrator or assisted live-in administrator has the skill to convince the elderly that she cannot be alone. You can find some of these people to help you to convince her.

Take care,

Nina