There a a few reasons why your wife can't get her feelings back for you..
#1.. There IS another man. It would be a big step forward for you to face that. People that "cling to hope" sometimes have a very hard time facing facts and the realities of what is REALLY going on. Denial takes over. You ARE in denial. The signs are all there. I have been at this for over 20 years and can't tell you how many times I have seen men such as you be in denial. You can't really move forward without all the facts or until you face the facts. What your wife has been doing is classic. It isn't new. It isn't different.
#2... Since she IS having an affair or interest in another man, then she views YOU as weak. She thinks she has given you every "hint" in the book and you still won't let go. This causes her feelings for you to be more of feeling sorry for you than respect. She would respect you more (which helps a woman to love a man) if you would show her that YOU GET IT. She doesn't want you. When a man shows a woman that he "gets it", then the dynamics in the relationship start to change. You are showing her you don't get it. She can see right through your "trying." It comes across as desperate. Almost as if you are begging for her to fall back in love with you if only you do the "right" things. That is NOT working for you. The reason again goes back to the fact that she has given her thoughts and heart to someone else. If that wasn't the case, then she would have NO reason to not give you a chance. You need to understand this dynamic because it is TRUE.
3# Asking her out is NOT going to work. She is involved with someone else. She doesn't want to hurt you. She doesn't feel you can emotionally handle it. (Can you?) Women are NOT attracted to men who seem emotionally weak. So, again, all the things that you are doing that you hope and think are working, are actually NOT working. Matter of fact, they are working AGAINST you because you aren't showing the natural male qualities that women are attracted to. It is one thing to be there for a woman when she WANTS to be with you and freely gives her love to you, but it works the opposite to be there for a woman that is giving you every sign she can without being mean, by trying to "nice guy" her into loving you. THAT does NOT work.
Your answer is to first realize and do what is necessary for you to realize and know she has her sights on another man. This would help you to understand why she is doing what she is doing. Facing that issue is terribly important. Denial is holding YOU back. Please stop the denial. Women don't just suddenly want out and then not try again when the man makes changes UNLESS there is someone else in the picture or unless there is some terrible abuse that has gone on. AND even when there has been terrible abuse, many times the woman STILL gives the man a "second chance" if there is no other man..
I would hope this doesn't hurt you, but it is such a waste of time to watch a guy like you continue to go on and on and on focusing on the wrong things and the wrong reasons only to be let down in the end...
Get a detective, snoop, whatever.. Find out the TRUTH. First and foremost...