Like the first poster said (sadly), you can't.
The alcoholic / addict will only change when they see more value in being "clean" that being drunk.
My father was an alcoholic for forty years. Running his family through the wringer didn't change him. Losing his job and having trouble professionally didn't change him. Losing all his friends didn't change him.
(As far as his record goes, he had three DUIs altogether and spent 30 days in jail two years ago for the last one. He was on probation with his driving and had his car "locked" with a breath-tester; the car still does not work right, although the device was eventually removed.)
He just kept drinking more and more until he was completely and perpetually inebriated. While he had been forced to attend AA meetings for quite awhile, he had a rotten attitude and kept pushing people away.
Last fall, he came literally within an inch of death from the drinking and barely made it alive from the hospital. (No kidding. I was shocked and amazed he pulled through, I fully expected to be planning a funeral within the week. He actually stopped breathing when I was with him and had to be "coded.") As it is, his liver is permanently damaged and he needs to be on meds for the rest of his natural lifespan.
Still, apparently the scare of that, coupled with the "discipline" of the hospital environment, was what he needed to want to go clean and stay clean.
He has been clean and sober since then, and there have been some startling and complete changes in his attitudes that suggest the change will be permanent rather than just temporary.
But look what it took. He just didn't want to get better, until he had to face his own death, and then he decided that was it.
What I think? Your mom hasn't lost her mind. Yes, for someone to choose what she is choosing seems completely illogical and insane. But she probably feels "locked" into her drinking and doesn't know how to cope with life any other way. Even if she'd try, she doesn't have the support group or knowledge to succeed (in her mind); and it's just too hard to stick with over the long haul.
I can understand why you're angry at her. (I was angry at my father at all the lies and senseless pain.) At the same time, I can empathize with her and know how hard it will be for her to move out of this place in her life, without a great deal of help.
All you (and others who care about her) can do is be there and be willing to help, if and when she decides it's time to change; and otherwise be willing to let her REALLY hit rock-bottom because she isn't ready to change yet. Keep the communication doors open, without letting her abuse you or use you.
As my father almost did, she might not pull out in time. I'm sorry you have to go through watching that happen. It's hard to know what sort of kindness on your part will "enable" her (bad) vs. being a show of honest compassion that will help her change (good).
Please feel free to contact me via my Yahoo profile page, if you have any questions or just need to dump. You might find some support with Ala-Teen or other similar orgs... if simply to have people around whom you know understand you.