Jez, do you have some sort of child protective agency in your area? What I'm thinking is that this child is definitely being abused by having her life endangered.
Perhaps at the very least, they could check out the situation and arrange for some sort of therapy or intervention for your mother. They might also find that the situation was truly problematic and remove the child.
I do know how difficult this is to deal with. My sister was also an alcoholic (she recently committed suicide, and I'm sure the alcoholism played a significant part in this). At one point, I tried to get the authorities involved in behalf of my nephew, although he was much older at the time. My sister was such a good liar, however, that I never got very far. It's worth a shot, though.
As far as giving her a "chance" is concerned, however, the point to remember is that unless your mother is somehow forced to face the reality of what she is doing, she will never change her behavior. Excusing it is NOT the answer. Unfortunately, it may take a serious accident to cause her to wake up and see what she is doing.
I fully empathize with your concern and frustration. I wish I could give you a really concrete answer, but I can't. I WILL keep you in my prayers, however. ****HUGS****