Some alcoholics cheat and some do not just as some non-alcoholics cheat and some do not. I don't buy the excuse that this is because she is drinking. Is she drunk at work everyday? Doesn't sound like it to me.
Please don't give this behavior a hall pass based on her alcoholism.
She sounds very depressed to me. The alcohol only makes it worse.
For me cheating is a non-issue. I put up with a relapse I tell you if I EVER found out my husband cheated on me it would be over. I have no desire whatsoever to repair that damage, and I would never trust him again anyway.
I am of the opinion that her decision to stay is another red flag. Sorry to have to say that, but when someone cheats and they want to repair the damage they don't make decisions like seeing that person everyday. They shouldn't want too, know what I mean? She should be kissing your a** and being as transparent as possible so that you can recover from this.