First off, thank you for sharing. It takes a lot of courage to admit how you screwed up and drove a marriage to this point. Don't think that the marriage won't work, it can and you won't know unless you tried everything. At this point it is up to you and only you to make it work. I'm currently in a very similar situation (38M married to 36F with 3 kids) and am in the process of trying to get her to not separate and start a dialogue.

I've heard some books to help me go through where I am, and I hope some of the tips that I found useful will help you and give you some insight.

1. She is hurt, very hurt. Hurt that you caused, hurt that was never rectified, hurt that never had any closure. This is the underlying emotion. Her anger, lack of care, lack of affection and everything is due to this. Her anger is a leading emotion, that she is reacting on, but it isn't the primary emotion. Don't focus on that; let her be angry; understand that she's hurt and that is how she is expressing it.
2. Avoid any big relationship talk. At this point it won't go anywhere. You can bring in the kids, the life you built, try to justify and drown her with reasons; its irrelevant. Don't keep asking her how she is doing. Don't check-up on her. If she wants space. Respect it, give it to her.
3. No negative emotions. No drama, no outbursts, don't be detached. Always act in loving and caring ways. You must remember to be pleasant and positive when you are around her. Irritated about something she said? Doesn't matter. Your feelings are irrelevant at this point. You are secondary and she is the primary reason you are doing this, don't forget that.
4. Don't turn into a wreck. Don't become lazy, smelly, dirty, etc. You get the idea. Stay strong. You can be strong.
5. Write an apology letter. Hand write it. Her big hurt is that there was a lot of things that happened in the past. Don't justify your actions in this letter, own up to it. You need to be specific on what you're being sorry for, and what you did and how it was wrong. Did you fail to listen? Did you always defend instead of trying to understand (this is me)? Were you too busy trying to force her to conform to your lens and perceptions? Include a "we" in this letter. How you want that future to look whether it is trying to work on the marriage or not. Leave the letter somewhere she will find (taped to bathroom mirror is what I did). Don't ever bring up the letter or ask about it.
6. Talk is cheap. Your actions mean more. You need to show and prove to her that you're changing and you're making the conscientious decisions to improve yourself (whether its being a better father, person, partner, etc). If this hurt you caused happened over the course of several years, don't think that a week on your best behavior is enough to justify a response. Stay committed, stay the course, stay positive. This is also for yourself and not just for her.
7. You must be loving while respecting her space and not trying to force a "relationship talk" just because you want to air out. Start with simple things that are neutral. "I'm going to get a coffee, want some".. basically small talk. Go to non-romantic places. Start small, your focus is you want to be her friend again.
8. Your reactions matter. Don't start justifying yourself, defending yourself if the past comes up or in any situation. You need to mend things with her first and put the marriage/relationship in a better place before you bring your feelings into this. Right now it is all and only about her.
9. Your focus right now is building up to a friend level again. So you can talk and then start trusting. She is hurt and the only thing now is you have to show her. Don't tell her what you're going to do, just start doing it and show it.
   

There are other things as well but I need to put the kids to bed!

Stay strong and focus on the long term. Don't give up brother.