In the same post, you're talking about both: A) Exposing Wife's misdeeds on facebook for all to see, in order to hurt her. B) Giving wife a heartfelt letter of apology, in the hopes it will pave the way towards reconciliation.
There's obviously a lot of inherent tension in those two options, and by "tension" I mean "WTF?!" But honestly, I get it. I think those competing goals of wanting to make WW feel the pain they made us feel, and wanting to just find a way past the pain together, are very natural and expected. I've been there, and still get tempted from time to time to act out.
So I understand the compulsions there, but really either of those would likely be a bad idea at this point.
There are definitely different sources that take different stances on exposure, and whether and how to go about it, but you have to be careful about moving quickly between different game plans. Not all sources will be compatible with each other, and mixing it up willy nilly may not take you closer to your goals.
Regarding the letter of apology, I would propose a different possibility to consider. One thing my phone coach had me do was to put myself in my wife's shoes and try to imagine how she thinks I see her, and what I did that was most painful to her. Once I identified the biggies, then she suggested I keep an eye out for either of those to be brought up by my wife, and take the opportunity to validate/empathize.
She had me write an apology letter to help me crystalize my thoughts, but it was very much for me only. WW is not going to be interested in a long canned apology and bearing of feelings, no matter how sincere it may be from your end.
In my case, one of the painful things I had done was to make my wife feel lonely by putting computer-time first, and often be dinking around on my phone/tablet even when we were hanging out together.
It took a while, but at one point when we were together at an event and I was being particularly "present" (chatting with people, active listening, etc) she brought it up later that night how different it was to feel like she wasn't the only one having to drive conversation for once. I responded with my validation/empathizing and integrated a thought or two from my apology letter, voicing how painful it must have been for her to feel alone even when we were in the same room together because I was spacing out, or on my electronics.
It was similar content as from my letter, but keeping it very brief and delivering it in the context of validating her comments... I could tell it really resonated with her. She said she felt "for the first time like I really got it." If I had just nailed her with the apology out of the blue, or given her a formal letter bearing my heart, it would have probably been dismissed out of hand.
No idea where my situation will ultimately end up, but for what it's worth I would strongly suggest you stay away from any letter-based dramatics of any sort.
Me, WW - Upper 30s BD - Apr 1 2016 EA - Apr 7 2016 (discovered; ongoing for months; did not confront right away) Confronted wife about EA - May 17 Wife sent NC email to OM - July 11