Welcome Hurt. Know that we have been the people out there who have been in the same trenches you are right now. Know that we have made mistakes, and we have learned from them. Know that when we tell you to do something, and you see it over and over that we know what we are talking about.
Your wife - Is a repeat offender. You were gracious enough to forgive and rugsweep the event and stay the previous time. Now this time it is really up to you to decide if she is worth the work of R. If she is then you absolutely have to lay down some ground rules. She will call them rules, she will say you are invading her privacy, and a million other ridiculous things. Don't listen. Her words mean nothing at this point. She has managed to lie to you and hid the truth effectively. Stop listening to her words, and start watching her actions. That is where it counts.
Your M - To have a healthy successful M your wife has to realize the pain she has caused, and figure out why she reached out to someone other than her spouse when life gets hard. She has to figure out her "why" and heal that within herself. If she doesn't then you will most assuredly go through this again, and again. How much soul crushing one person can take? That's up to you.
YOU- You and your kids HAVE to be your priority now, and as long as she "cant decide" and take her time to "think" you HAVE to protect your heart, and your children from her. This isn't a good mom, this isn't a good wife. This is someone who is equal to a drug addict in their thoughts and actions. It becomes an addiction, and impairs her ability to think, and make good decisions.
That being said you need to get thee to an Attorney promptly. D or R, it doesn't matter, you need to understand how this would play out, and what would most likely happen should she choose to leave. Don't operate from a place of fear and assumption. It often leads to horrible choices.
You also need to get thee to the Dr pronto. Get STD tested. This is the icky reality of the situation, and do NOT be intimate with your wife, without protection until she is proven NC with her AP, and has a clean bill of health from her Dr. If she says she used protection, do NOT believe her. Rarely in the land of Skittle shitting Unicorns is protection needed. She is a proven liar and cheater. Do NOT risk your, and your childrens health on this.
This is tough stuff, we get that. Read a few profiles, you will find those of us that have R'd successfully have a common theme. We all quit playing the nice guy, and demanded the love and respect we deserved. We all reached a point where we were willing to loose our M to save it.
Lastly, I see you accepting blame for her choices. Please do not do this. It is wrong, and it is unhealthy. Her choices belong to no one but her. Your M may have been hectic, and you two may have lost some of the bond, and romance, but this does not EVER justify cheating. It does justify spending more time and energy on each other.
You are stronger than you ever imagined you could be, you are smart, you are capable. Do not accept less than you deserve. Do not accept less than what you would want for your own progeny.
Stand up for you, your M, and your kids, and demand the love, respect, and honesty you deserve.