Your situation is one of the most painful events in a marriage. If I understand you correctly, you are accepting 100% of the responsibility for your wife’s affair. Your wife blames you, your parents in law blame you and you blame yourself. Stop right there. There’s an awful lot of blame being thrown around here.
It’s very noble that you accept responsibility for creating a situation that made your wife unhappy. So unhappy that felt the need to run into arms of another man and despite her affair being discovered, she is carrying forward with it, while presumably living in the matrimonial home with you.
I truly understand your pain and confusion but you have to accept that the entire dynamics of your marriage have undergone a profound change. Waiting at home hoping that one day guilt will stop her having the affair and returning to your arms is no solution. Even if her affair “ended”, the trust is broken. Some marriages can be rebuilt after an affair but those that are is usually when the unfaithful partner apologises, seeks forgiveness and takes positive steps to rebuild the connections. Your letter did not convey that any of things are likely to happen.
Your first obligation to devise a plan that ensures your children are looked after properly in a stable and loving environment. If that involves separating from your wife, then so be it.
By all means seek counselling to ensure that the separation is as smooth as possible and that adequate provisions for the children are given top priority. I know this is probably not what you want to hear but if your marriage has any chance of surviving, you must change the status quo.
Good luck and don’t hesitate to respond with posts if you need support, advice or just to have a rant.