Sorry you are here Derek and the situation you find yourself in. Your life has been turned upside down by your wife's actions. You are only a couple of months out...way too soon, you should have posted in Just Found Out... way too soon for Reconciliation even if that is what you want.
You need to realize that you have been playing the pick-me-dance with your WW. This never works. You can't "nice" your wife out of her affair. This is much more than a simple infatuation if she has slept with him... 2 times, at least and AS FAR AS YOU KNOW? Keep digging there's usually more... is this her first affair? Or just the first affair that you know of?
She currently has you reacting to her actions, needs and emotions. You need to flip this around. Start taking action that makes your WW react to you. Things you must demand of your WW:
-She needs to know that she has lost your trust for the present and probably for sometime in the future. To show her how much trust is lost, you need to ask her whether her four children are truly yours (even if you have no doubt that the kids are yours) She needs to pay for DNA testing for all 4 kids. You need to do the testing yourself and have the results sent directly to you at work (If kids ask why, just tell them you're doing an ancestry test)
-She needs to give you a timeline of her affair in WRITING, give her an ultimatum for when it will be done, everything must be in it... not just acts but how she was feeling and thinking during her affair. Tell her after the timeline is done, you will seek a polygraph test to determine how truthful her timeline really is.
-You and your WW need to have STD testing done IMMEDIATELY. Doesn't matter if they used protection (what does she say?) Protect your health and demand the STD testing and that you need to have the results directly from the lab/doctor's office.
-You will need all passwords to her phone, computer and tablets as well as username/passwords to all her social media accounts as well as personal and work email... no exceptions. Start searching for a burner phone in the house.
-Consider getting a couple/several VARs (Voice Activated Recorders) Place one in her vehicle (obviously don't tell her you're doing this) You need to know if she is still in contact with AP and also what is she discussing with her family and friends. Does she talk on the phone in a certain room in the house... put a VAR there. Also keep one on your person at all times to record your interactions with your WW.
-Read up on the 180/gray rock immediately and implement it hard.
-Have a consult with an attorney, lots of attorneys have free initial consults. You need to see what a divorce entails in your state or country, what the laws are regarding custody, child support and alimony, how long it will take and approximately how much it will cost you. Tell your wife -afterwards- that you have consulted with an attorney. YOUR MARRIAGE IS NOT A DRESS REHEARSAL... this is real life with real consequences.
-Finally, Derek, do you have friends or family that are close by and you can stay with for a few days? If not, can you afford a hotel/motel/AirBNB for a few days? You need some physical space from your WW for a time period. Go no contact with your WW for a few days, don't tell her where you are other than you will be back when you leave the house. You WW begged you not to divorce but still stayed in contact with her AP after confrontation. She needs to have a serious taste of being a single mom for a time. Let her deal with the kids alone for a few days. While away, take some time to think what you want your marriage to be and how do you want to live...you need to be away from everything so you can have some clarity to think what you want going forward. Yes, it will suck to be away from the kids for a few days... but if you divorce this will be your reality, might as well see what it will feel like beforehand. Your wife's "fantasy" needs to be blown up... do it ASAP.
Don't be a doormat Derek. You will regret it in the years ahead, your kids deserve better (they've also been betrayed). Stop the pick-me-dance. You are the prize, your wife is now the villain in your marriage. Start treating her as one.
Start taking action. Good luck and let us know how it's going.