I see your dilemma clearly.
Ultimately when it comes to tolerance for our mistakes it is the other persons values that determine the outcome of your situation
However, guilt is hurtful. Lets be realistic here. All people arent perfect. We at times cross boundaries and, depending on to what extent, we can sometimes only acknowledge the severity AFTER we turn back.
Your guilt level about this action tells me you are unlikely to go down that path again. So,, what is reasonable when it comes to forgiveness? Well, if your wife believes it is unforgivable there is little you can do.
You cannot erase the past. If you believe you should be forgiven and you'd never do it again, then that right of thought/belief, is as important as your wifes right to not forgive you.
Finally, there is a point to where issues like these can ruin your life eg lose your marriage and full time parenthood . If that happens then I'd suggest your wifes unforgiving stance is not so reasonable and there is more to it...if it took an emotional affair to break a family up then there is something else that isnt right.
In that case, dont beat yourself up about it. Hold onto your dignity and accept you messed it up but also be fair to yourself and accept the marriage had its flaws and lack of commitment on the other side.
If we didnt make mistakes we wouldnt be human, it is the extent of such mistakes that determine the end result and willingness to forgive.
When my marriage failed I was devastated but it wasnt the end of my world. You'll always be a dad and your importance to your children is paramount.
Mine happened in 1996. My kids were 7&4yo. I recently walked my eldest down the aisle. No one else could have filled my shoes better than my own feet...