If you or your wife cheated would you stay together for the kids?

I would run not walk out of this relationship. Get yourself a lawyer and ask for joint custody of the children. You cannot live like this and put up with mental abuse. You will be in a much better mind state making the best out of your quality time with the kids without her here in your face. She is not worth tolerating this horrible situation so hold on to your dignity and get out. This woman has no morals or respect for you and you can only change the way you feel by looking at it differently. Find yourself someone in the future who will and can be there for you and your children sweetie.