I wish I could give you some positive feedback , but unfortunately it doesn't look that bright. That said, some things are blessings in disguise.
Although I work in the psychiatric field (not psychology ) , I dare to say that your wife has a major personality disorder...and she is possibly a sociopath. Please, don't take offence to what I say. I am talking both from professional and personal experiences .
First of all, in plain English , your wife treats you like a dirt-bag and has no respect for you. Whether she is cheating or not( is one thing ). She is treating you with such disrespect and that is worth of walking away from her for that reason only.
The million dollar question is; can you accept her as she is? With that sort of behaviour ? It sounds like she is having a life of her own, doesn't want your affection and love and couldn't care less what you thinking of her. I'm really sorry that things are the way they are as it sounds that you really love her .
My advice is - Be super firm! Ger proof that she is cheating ( cause then you can legally divorce her and have the kids ), and then put the divorce papers on the table, no emotions, very cool, very calm, very silent. The secret to do this is to prepare! Mostly to prepare emotionally . You need to accept that this is not the person you thought she was. She is a selfish creature that has no care for you and would probably traumatise your kids as well (with her behaviour) by exposing them to this madness.
You're not alone my friend, so many people going through these sort of situations. Think of your kids and your self only! Detouch yourself from her, for she is damaging you slowly( perhaps even unintentionally ).
My guess is , if you do it as calm and calculative, she will beg you in the end. But she might not.
Think of your children, because I believe that this current situation will hurt them more than a divorce . Plus you have to lead by example! You show them how to live a life free of any tyrannic creature that enjoys treating you like used condom .