The No Contact letter is only important if your wife is willing to end the affair.
Your situation has not progressed to the point of sending a no contact letter, however, it is very telling that she refused to send one, and also that she refused to name her lover - both huge red flags that she had no intention of ending the affair.
You made some mistakes in handling your wife's affair. The biggest one was in not demanding to know who the other man is. So you have to go back to step 1 - find out who the other man is. Get the details of the affair. Determine if it still is going on. Continue to investigate.
In the meantime, ask your wife to tell you the truth. Tell your wife that you are too tired and worn out to snoop or spy on her and you have no intention of doing so (even though you really are). Tell her you expect her to tell you the truth on her own, or you will divorce her. Tell her to start with the other man's name, address, phone number, email, etc.
Tell your wife that you cannot control her. You can only control yourself and how you react to what she does or doesn't do, what you are willing to accept in your marriage and what you are not willing to accept in your marriage.
Tell your wife that if she doesn't tell you any detail you ask her for about the affair, you will file for divorce. Then, if she refuses, do it. Really, if she would choose to keep these secrets over your marriage, what have you lost? Also, divorce is a long process that can be canceled if your wife does agree to your conditions.
Tell her these are your conditions for staying married to her:
1. The truth. Polygraph if you need it.
2. Transparency. Access to all devices and accounts. No deleting of anything going forward. She let's you know of her whereabouts 24/7. Get rid of facebook or downgrade to a cell phone with very limited features if you determine that's what you need.
3. No contact letter to the other man stating how horribly ashamed she is of her behavior, how terrible she feels for having risked losing her family, marriage, and husband, which mean more than anything in the world to her, that other man is never attempt to contact her again and if he does she will file harassment charges against him. AGAIN, THERE IS NO POINT IN THIS NO CONTACT LETTER IF SHE HASN'T DEMONSTRATED (BY ACTIONS, NOT JUST WORDS) HER DESIRE TO STAY IN THE MARRIAGE. Telling you the truth would be one way to demonstrate this, transparency would be another, doing what you reasonably ask her to do in order for you to heal is another.
4. Any other conditions you need to help you heal from this. There may be additional conditions based on the truth of the affair.
You may not be able to save your marriage no matter what you do. You have to be willing to lose your marriage in order to save it. Your wife is playing a game of chicken with you. She is counting on the fact that you are too afraid of breaking up the family to file for divorce.
Also, your wife cheated on you with him for three years. If she could have, she would have left you for him already. There is a reason that she hasn't left you for him. Either he is married, financially unstable, etc. In any event, my bet is that she doesn't want to divorce any more than you do - she is just bolder in being able to risk it, feels comfortable you won't do it, or feels comfortable she can manipulate you out of it.
PREPARE YOURSELF FOR THE WORST: If you tell the story you told here to any normal person, they are going to tell you that your wife probably still is cheating on you.