First off I am really sorry you had to have that conversation with your wife.
Think about your response to her discussing starting the process of divorce. **I will continue to support her financially and emotionally***
Why would you say that when she is starting the D process and is having an affair? The reason is that you are grasping to re-establish a feeling of control over your life.
As quoted by Accuracy: Your brain has convinced itself that getting W back, or getting W to apologize and declare a desire to have you back is the very best and fastest way to restore your feeling of being in control.
This is why detachment is so important. When you are completely detached your response would be " Honey, I am really sorry you are starting the D process, I would love a chance to work on our relationship, I hope you find the happiness you are searching for in your new relationship". Then you Lawyer up.
Then you start to move on in your life, if she contacts you in the future about reconciliation you can make a decision at that time.
For the most part you seem like a strong centered dude who will be an awesome role model for his boys.
I still think she may come around but you have to act strong and stop the I'll be there for you BS and it may be years down the road.