I'm going to start with what I feel the issue with the framing of the question.
"what possessed you to leave, throw away your family for someone you knew for four weeks?"
Ain't nothing possessed you. There was no spirit from another dimension or universe that came in and took over your body. If you are spiritual and believe in those things, that is fine, but using that language shifts responsibility away from you and over to whatever possesses you, which is wrong. A better way to frame the question, "Why did you justify (to yourself) that throwing away your family for a near stranger was the right choice?"
I carefully selected those words because you did make a conscious choice to have the affair and you made hurtful choices to continue that affair and to leave your family. The answer to that question is really something that you will have to work on with a good therapist and dig deep within yourself to understand. The AP could've been anyone who batted her eyes at you, she just happened to be the one who made herself most available to you at the moment you wanted her to be.
I think what your BW is getting is more how can she even consider herself safe in a relationship with you again. You've been
married long enough to have three kids, so there are years of history and marriage that you with the wave of a wand threw out. Your wife was looking to you to be a partner and parent to your children. I mean, after all, they are your children too and because the AP invited you to move in with her and abandon all of your responsibilities and obligations to her and the kids. Now you stand before her asking her to reconcile and let you back into her life when you've proven yourself quite unreliable in that regard.
I hope what you recognize and what I'm sure she recognizes is that your AP was a total fantasy and I'm suspecting that you figured that out pretty quickly. It's one thing to be casually sleeping with someone and seeing them for an illicit affair, but it is another to be married to them, owning a home, paying bills, running the kids to soccer practice and living in the real world. Of course your marriage could never stack up to the fantasy that you were living with the AP. I mean, when you are in the affair, you don't ever have to deal with the messy life that the AP has. If your complaint is that your wife leaves her hair in the shower drain or leaves her dirty dishes in the sink for you to clean up, well, in an affair, you don't have to deal with those aspects of your AP, so by comparison, everything is coming up sunshine and rainbows for you. That is, until you move in with the AP find out that just like everyone else, she has her gross quirks that you find as equally revolting. By the way, I'm just using the hair in the drain as an example, but I think you get the point. If you've lived with someone long enough, there are little things they do from time to time that annoy you, lord knows my wife would say the same about me.
The point is that it was all in your head. You had convinced yourself that the AP was the super bestest thing that ever happened to you because unlike your BS, the AP didn't ask you to take out the trash or go help the kids with homework, she wanted nothing from you other than ego kibbles and sex, which if we are being honest were your requirements from that relationship.
If I'm sitting your BW's shoes today, I'm looking at this and saying to myself, sure, tinytim is now coming to his senses and he has come home, but he has also proven that his commitment to me and his children is conditional. I mean, in four weeks of knowing your AP, you convinced yourself to throw away a long term relationship and your family all for some ego kibbles and sex. The fundamental issue that she is wrestling with is how committed really are you and can she trust you to be there for her and the kids. Whatever justifications you made to yourself as to why it was okay to have an affair need to be examined. You need to really think about them and address each of them. Because if you do not, you are bound to repeat the same behaviors down the road. Your wife needs to understand that the next time you guys have a disagreement or fight about something that you won't just leave an run off to AP's house or find another AP. You will have to demonstrate to her through your actions, not words, that you are committed to her and the kids before she will have any level of trust in you. Keep in mind that for some people infidelity is a deal breaker, and so it is also possible that no matter what you do to move heaven and earth to show your wife you do truly love her and only want her, it may never be enough. That is a tough pill for a wayward to swallow, but it is the honest truth.