I know this might sound a little ass-backward to you, but I advise that you don't try to do anything in terms of saying you are sorry or even trying to open up to her. I say this for two reasons. First, that ship has sailed. Words mean nothing, especially when we say them and then do the opposite. You've been saying sorry for over a decade, nothing has changed as a result of it, so just stop. A hollow apology is worse than no apology. Stop saying you are sorry and just start doing all the things you've promised. That will mean much more to her, promise.
The second reason is that you are still trying to obtain your own desired outcome, your own goal. Saying you are sorry is about how YOU feel. It has value to YOU, it has none to her. Wanting things to work out between you is what YOU want. She's been talking about divorce.
Here is the thing to understand. For the past 14 (or however many) years, you've been consumed by what you need, and what you want, and when someone has to lose, it's always her. Every time you get what you want, she gets hurt. Every time you need something, you find a way to get it, even if it takes lying and hiding and all sorts of effort. But when it comes to what she needs, all that effort and concern goes away. Maybe you feel that you are taking care of your own needs and she can take care of hers, but do you honestly make her feel loved and appreciated? If she isn't being loved and appreciated then the hell would she want to be closer to you? Again, I don't say these things to be mean... I've been in your shoes, and thought and felt the same way you do. This is just what I see and I am relaying it to you, like an announcer at a football game, commenting on the play-by-play, the player stats, and where the team is strong and weak.
Look, if you were building a house, and started to try and work on the roof before the foundation was even poured... well, if you saw someone doing that, what advice would you give them? Would you tell them that building a roof is useless when a foundation, utils, walls, doors and everything else is needed first?
You guys having a relationship is a roof. You not feeling like shit is a satellite. So let's start with the foundation, okay?
Here's an idea for you. It will give you lots of safe stuff to talk about and won't require you to come up with topics, at all. Remember that Brene Brown book I mentioned? Did you get a copy? If you can't afford one let me know I'll buy you a copy. Then, make a deal with your wife (if she's willing). The deal is simply this. Some time each evening, perhaps before bed is a good time, take 15-30 minutes and take turns reading the book to each other. Some chapters are short and can be read in that time, others you may have to break up over several days. Either way, when you are done reading (or when you read something interesting and the mood strikes you) TALK about what you read, what she meant, what you took from it. That's all. Other than the cost of the book it costs nothing, you can make your own schedule, you can make your own rules, just the two of you. The great thing about this is that it will give you that "in" to talk that you asked for, and while I can't speak for her, my guess is that she would be really happy to have a conversation with you where she doesn't have to hate you and you don't have be either defensive or angry or shame filled. I recommend this from experience. My wife and I did this, and while it didn't fix everything, it did help us to learn to talk and communicate and share in a very easy and pleasant way.
I take it that you don't like to talk much, or write much, or read books, and I get that. Everyone is different. This is one of those things however where certain efforts need to be made. A football player who doesn't want to exercise or practice isn't going to get far. If you wanna play, you gotta practice. If you want to build a relationship, you need to talk and interact.
If you want other safe things to do, just ask. She likes to cook, right? Would you help her cook a meal? Pick a recipe together and then try to make it as a team. Play a board game together. Go take a walk together, you don't even have to talk. Ask her to join you in a project you both always put off for later. Go fix all those loose doors in the stable or whatever needs doing. The point is to show some interest in HER, in doing things with her, spending time with her. And if she says no, she doesn't want to do those things... then that needs to be okay. Don't get mad, unless its yourself you are getting mad at. Things are this hard and suck this much because of your efforts so far, so that. Then you go work on yourself. See the IC, join a support group, and if you can't do those things, SI is free and honestly, it's sort of an aggregated support system, you'll get advice here that money can't buy, from people who have actually gone through this, and who actually care.
Oh, and while your kids have no reason to trust or believe you either... they probably DO need to hear from you. No excuses, no justification, no minimizing. Just say, "I've done a lot of things I'm not proud of, and I know that in doing so, I hurt you, and didn't give you the love, guidance and protection that I should have. I did not put you first. I have no excuse, I just want you to know that you deserved better from me, and that's on me. It's not a reflection of your value as a person. I want to be a better person and I'm taking steps to do so. You make me want to be a better person. That's all I have to say for now, but if you want to tell me how you feel, I'll promise to sit and listen." And then you shut up and let them go off on you in whatever way they need to, because the goal is to let them get their feelings out. You can worry about how you feel about things later, just listen to them, and believe what they say to you.
FWIW, it was really, really nice to see you post here again today. While I can feel how uncomfortable it is for you, you are still making the effort, and that's all that's being asked of you right now. Just keep trying. You've made a lot of mistakes, but you've always accomplished some really wonderful things in your life. Look at the great kids you made if nothing else. Look at some of the rare skills you have. My gut tells me that you suffered through some rough shit in life, probably in your youth, and whatever that was, it left you feeling forgotten and unwanted and unimportant, and so you live a life that reflects that crappy view of yourself. Don't believe it, not for a second. Even after hurting everyone, your wife and kids... they still love you. They also hate you, yes, but for some reason, they also still love you, and you need to figure out why, because whatever it is, it is something good and wonderful. Otherwise they'd have ditched your ass long ago. Your kids want and need a dad that makes them feel important, appreciated, protected and loved. Go be that man and I promise you will love yourself as a result.
Keep coming back.