This is a lot. I'm sorry you are going through this. However, this situation has occurred because you and Shaun have spent decades enabling Kelly's drinking. This situation isn't particularly complex when framed that way.
Kelly is an alcoholic and she has needed help since at least 2009. You and Shaun have both glossed over this, dismissing it as, "Oh, Kelly was lucky to avoid a DUI back then," instead of being alarmed that she drove drunk (endangering others, as well as herself) and taking it as a call for needed action.
Shaun spent his marriage picking fights with Kelly about her hobbies, and avoiding going to serious therapy (either for himself or both of them), rather than addressing his wife's alcoholism. It sounds like he never went to an AA meeting himself (or similar support group) or attempted to get Kelly the help she needed.
When Kelly finally got a DUI for real and lost being able to drive completely, again Shaun did not address the underlying alcoholism. He could have told Kelly he was leaving if she didn't go into rehab, for example. Instead he just got mad every time she Ubered home drunk, as if the issue were her staying out late and partying, rather than having a serious underlying addiction that requires a serious mental health intervention. (Staying out late and partying would be the symptoms, not the cause.)
Meanwhile, Shaun seemingly decided to mostly ignore Kelly while continuing to live with her, while moving on emotionally with Mel, deciding to have a child with Mel, and "buying Mel a ring." What does "buying Mel a ring" mean when Shaun is legally married to Kelly? Did he ever have a talk with Kelly about whether they wanted to remain married or not, or about creating some type of arrangement where Shaun has marriage-like entanglements with both Kelly and Mel?
You say Mel hates Kelly. And Kelly seemingly has been miserable for years watching Shaun and Mel be together in her own house while she and Shaun no longer have a sexual relationship. Why did Shaun just passively carry on with a situation where his alcoholic wife is miserable and his girlfriend hates his wife? Why decide to bring a child into that household? Why not just divorce Kelly years ago?
To me, it seems like Shaun just decided to ignore Kelly and maybe hope she would go away. (Buying a ring for Mel and not directly talking about the implications with Kelly seems like he just wanted to push Kelly to leave him without him having to initiate it.)
And you... you claim to be surprised at learning the true extent of Kelly's alcoholism, but you've known her for decades... She's spent years avoiding DUIs and then getting a DUI. Heathy people don't just get DUIs.
This reminds me so much of my ex's friend group, a subculture based completely around recreational drug use. Many people in the group had serious issues with substance abuse. Yet the dynamics of their behavior was such that everyone else claimed to be surprised when it turned out someone wasn't just "fun," but had a serious problem that was destroying their life and relationships.
I literally heard people say, "But I don't understand how this happened! I partied with her every weekend for years and she seemed totally fine!"

It's significant to me that your title to this thread mentions absolutely everything except the actual problem, the alcoholism.
Kelly's communication failures are directly related to her out-of-control drinking and underlying emotional issues.
Kelly's "cheating" behavior is merely a symptom of the fact that she is drunk all the time.
Also, I don't understand your emphasis on the "cheating," as if that is the biggest problem and the thing you need to talk to Shaun about. If Kelly and Shaun haven't been sexually intimate in years, why does she need his permission to fool around with other people?
I get that Shaun would have needed to give his permission for his wife to start dating his longtime best friend/best man in their wedding. I get that you didn't understand at the time how poorly Kelly had communicated with Shaun about the start of your relationship.
But I don't understand why Kelly fooling around with others is such a big problem. (It is the least of the problems going on here!)
I think I just don't understand swinger mentality, or the appeal of a closed quad. It just means that if one person is unhappy in the quad dynamic (or has a sexually dead marriage), they are trapped and don't have the freedom to find other poly relationships.
Swinging is very focused on preserving the original dyad couples. Most of your post is about Shaun and Kelly's relationship, instead of YOUR relationship with Kelly. What difference does it make if Shaun and Kelly have had a terrible marriage for years, other than indicating that Shaun and Kelly are both emotionally unhealthy and would prefer avoidance and denial to dealing with their issues?
Like, you seem very upset about Shaun and Kelly's marriage, rather than about what YOU are going to about YOUR girlfriend's serious alcohol problem. Some of that is because of the wonky boundaries that occur when you date your best friend's wife. You're stuck in the role of being unsure how to give your best friend advice about his marriage, because YOU are in a relationship with his wife.
It's not your problem to figure out what Shaun and Mel are going to do about the baby and their living situation. (Maybe Mel can move in with her mom until Shaun gets Kelly out of the house.) But it IS your problem to figure out what YOU are going to do about Kelly's alcoholism.
Like, if you want to help, could you do the legwork of finding a rehab facility, figuring out how it could be paid for, insurance, etc.? (I am hoping that when you mentioned Shaun "moved into the hospital" with Mel, it means that you're not in the US, but in a country with actual good healthcare.)
Then you can offer Kelly the choice: she goes into rehab or you walk.