People who blame mistakes on alcohol are usually losers, sorry to say it. Alcohol doesn't make you do things you don't want to do, in fact it's the opposite, it gives you liquid courage and liquid confidence to do the things you always wanted to do but were afraid to do. If she cheated when she was drunk, it's just because she wanted to cheat when she was sober. She did a good job though, she made you feel guilty for getting angry at her for cheating, LOL! Priceless, she cheats but you feel guilty, nice job! Again listen to yourself, you are "worried" that being too hard on her is going to backfire. Really?! What you've done so far hasn't worked for you, what would you have to lose?
Kick her out.
People only realize what they had when they lose it.
She will only realize your loss when she has lost you.
Let her go, have her move out, don't worry about her not having friends, she found enough friends to fool around with and I'm sure she has had more than sexual partner since this has all happened, don't kid yourself. When you have real walking power, the kind of attitude that says to your partner, you cheat on me, you lose me forever, that's when things turn around, that's when you will discover if they really want you or not and if they don't really want you enough to fight hard for you and show you they made a mistake and want to regain your trust again, well then... you really haven't lost anything worthwhile.
Your decision in the end but I'll remind you that what you have done so far hasn't worked at all.
"What is best for my kids is best for me"
Persevere = happily being patient over a long period of time