You are filing for divorce. Will your kids find out? Have they been told yet? If so, what happened?
It is not uncommon to hear about affairs after the kids have left the nest. Cheating itself is very common. The nature of it is for her to want both him and you, not lose either of you, hence the "cheating" - it is against the "rules" of a monogamous relationship.
So she lies about it. She is getting some needs met from him, some by you. She gets romance/sex from him, she gets stability/finances/repuation from you. Let's face it, she could have divorced you if she wanted to, but she didn't; she could have broken up with him if she wanted to, but she didn't; she has chosen to do whatever she can, lies or not, telling you she wants to work on the marriage, telling him she's has a "live-in separation" agreement with you, so she can keep both of you.
So now that you caught her, and she openly is cheating in your face, she seems to be manipulating you that she is "confused" and trying to drag this out as much as possible - to keep her "having her cake and eating it too" desire. It is kind of a very immature selfish mentality, she may even be resentful and angry of you not letting her do this. Her mentality in that respect is that of a pre-teen. Her mentality of the "love" with the other man is that of teenager.
Every affair I have heard of said that the cheater felt "alive" like they haven't since they were in high school.
It is an infatuation with the guy, and it is in what I refer to the "affair bubble," a kind of invisible bubble around them that keeps all reality out. No finances, no chores, no kids, just sex and I love you's. When you start to divorce her, when the kids and her family find out, when she has to start actually live with him day in and day out, the reality eventually will seep in and the "in love" infatuation will fade or burst. In the meantime, very likely, your give-a-shit will break before that, and after her being so cruel, you will not even want her back after a short while.
In the meantime, if you haven't told your wife's family and yours about the divorce and the reason for it, you should. Just tell them she has been having an affair and she refuses to end it. She said she would, and she didn't, and instead she ramped it up.
If your wife wants to work with you, truly, then let her do some actions to prove it to you. Let her willingly find another job, get tested for STDs, and tell her family and your kids that she is committing to you and ending the affair with him. She should be doing things to let you know that she is truthful, to the extent possible that she is no longer in contact with him. Like changing her phone/email, deleting social media, etc., until your marriage is on more stable ground.
But don't get your hopes up. And make sure she is doing the ACTIONS, not just words. She will tell you she is going to do it, but that is not good enough. She actually has to do it.