Yes, everyone on here is going to tell you to divorce this man.
But, ultimately it is what you want deep down and whether you can live with your decision. Also, can your children live with your decision, if you decide to fight for your marriage, as this affects them too.
As I see it you have 2 options.
You separate, divorce and start your life again.
You seek marriage counselling, have open conversations and try and fight for what you have left. Let him know it's not alright, it's not something you will tolerate again, let him know how betrayed and hurt he has made you feel and that this is all on him. Midlife crisis, whatever, we all go through it but it doesn't send us all running into the arms of someone else. He needs to be a grown up and face his problems, not try and replace them.
I am not condoning what he did, and yes he did a bad thing, but sounds like even if he had realised he made a mistake this woman was not going to let him go, I am picturing Glen Close whilst typing this (although an uglier, fatter, big busted version).
Ultimately, your life with this man is never going to be the same as it was, and there will always be an element of doubt. Who is that text from? Is he being more secretive with his phone? Should I check his emails? He said he would be 10 mins and he's been half an hour, where is he? Who is he with?
I wish you luck. But do what feels right to you, not what feels right to him, or your children or friends.