My quick answer to the question that you're really asking- is there hope for your marriage- would be "yes".
Unfortunately, your wife is still in the fog of the affair. Your wife has to completely cut off all communication with this guy. Reading one email, one phone call, one text- even a glance at a photo, can keep her sucked into the affair. Your wife should give you access to everything, email accounts, passwords, phone etc.. If she provides this, then she really is willing to work with you, despite the fact that she does not love you right now.
It will be extremely difficult for your wife to cut this man out of her life. With no contact, she will go into a withdrawel process of sorts- like an addict missing their drug of choice. Given enough time, she will emerge from this withdrawal, and only then will she be able to open herself up to feeling something for you. For my wife it took probably five months (if memory serves). You may come home to your wife in tears because she is missing her conversations with this other person. Be understanding, it will be very difficult for you both.
While she is in her withdrawel, continue to be the husband that you want to be in your marriage. While it won't make many deposits into her "love bank" , it will make some. It will also show her that your changes are real. Make no attempt at anything other than hugs or a kiss on the cheek. She will want no intimacy from you at this point. Take cold showers. During this period, I started leaving my wife little "love" notes each morning. Nothing fancy, just notes that told her how much I appreciated her, and what a great woman and mom I thought her to be. It's a practice I still keep up to this day, and something she tells me she appreciates very much in our relationship. She thought the notes would stop in days or weeks. I've kept it up for years.
Your wife does not believe she will ever feel love for you again. She certainly does not believe she will ever want to be intimate with you again. My wife was very frank and told me those things quite plainly after she broke off contact. I started to doubt it myself, honestly. The truth is, she had those feelings for you once upon a time, and if your changes are real, and she sees them, she will fall for you again. Unlike any other man on the face of the earth, you have a distinct advantage in winning back your wife- you are the father of her children.
Today, my marriage is the best it's ever been. My wife thanks me tearfully several times a year for fighting for her and saving our marriage. She says I'm the best thing that's ever happened to her and that I saved her from disaster. You could be it the same place too.
I caution you to watch out for false starts, especially since this is so new. When I had the first bomb drop on me, I thought my wife was resolute on recovery. I was wrong. The addiction was too much, and she just hid it better. That's why transparency and open communication is so imporatant. You have a window of opportunity here to make a start, but keep your eyes and ears open! I'm not trying to worry you, but it does not happen often that there is no relapse.
You can read my early posts if you want to see how my story went. I'm not a prolific poster, so there are holes in my story :-).
