((((((((((((((hugs)))))))))))))
Sending you a shot of courage hotpeppers.
She may well admit she has a problem. Admitting the problem can go on for years with no change in the actual problem!
It sounds like you have some boundaries and consequences lined up that will serve to restore YOUR sanity! Be strong, keep the kid's safety and all of your long-term mental health as your #1 priority (and they definitely know way more than you think - I sure did even though my mom chose never to directly discuss my dad's alcoholism with us), they will be relieved that you finally give it a name and reassure them you are the sober parent who will always be there to talk to them and care for them, and that it is not their fault and out of everyone's control except mom's. It is not easy - and just like you do, they love her, as they should, and it is just a very sucky hand to be dealt.
Peace- bluer skies ahead I hope!
B