I completely understand, OP, if you don’t end up leaving. This is entirely your decision. And it’s a big one. It’s not as easy as some make out - and I have been through something similar.
My DH eventually admitted, when confronted with my suspicions, to having cheated, seven years ago. Since the infidelity, we have had two more children and are very much happily married. I can attest to the fact that people can change.
There were lies, there was of course deceit, and the trust was obviously broken for a long time. And it was extremely difficult to move on from. However, I very rarely think about it now. And that is something I never ever thought I would say.
You have five children with this man. He has lied to you and deceived you and is continuing to do so. But I think you need to make the options clear to him. Continuing to lie will not help him, because you know in your heart what the truth is. He needs to tell you the truth and only then will you both ever have a chance of rebuilding trust over the years to come. I do hope for your sake and for your children’s sake that you are able to do this, as for me the alternative seemed far more painful.
This isn’t a case of what he deserves or vengeance against the two of them. This is about a turning point in your lives and which path you wish to take. Only you know which one you are emotionally and mentally prepared for. No one here can make that decision for you. If you are strong enough to leave him, I would admire you very much. If you are strong enough to stay with this man and eventually forgive him, then I would admire you for that as well. But the latter can only be done if he agrees to some very strict conditions - one being that he must tell you the entire truth, that he must cut all contact with the OW, and that he must allow you to get the family’s money back from her. He must then work incredibly hard to earn back your trust (this can take years).
But the point of my post is that it CAN be done, there may well be a light at the end of the tunnel in your marriage, as there was for mine. And I am thankful now that my DH was honest about what he had done, earned back my trust, that I chose to forgive him and that he is now a loving husband and wonderful father to our four children.
I wish you the very best of luck. There will be pain and a lot of tears but trust that whichever path you take, you will come out happy and stronger on the other side. 