I guess I can say 'been there, done that' too... and so has my new husband. However, even with lots of experience, every situation is different. After my unplanned baby was born I had a terrible few months where my boyfriend (of 2 1/2 years) at the time cheated on me and left me for the young girl he knocked up... leaving me with an almost 1 year old baby, an eviction notice, no job, and $8000 in immediate debt (he'd wracked up my credit card and drained my overdraft before he left). That's the short story.
The first thing I can say is your kids... they are THE most important thing... all your decisions should revolve around what is best for them. I completely agree with whoever said 'you can't work things out unless you can talk to eachother'. Turned out I couldn't have a decent conversation with my ex no matter how I tried... he always got his back up and yelled and threatened me. So I ended up raising my son and paying for everything myself.
Every relationship has gone downhill mostly because 2 people both made poor decisions (sometimes one more than the other... but you can always find something on both sides). It's never one persons fault. So, step one would be to take a good look at your relationship and find the things that you could have done differently. If you can't think of anything then you weren't looking, you weren't listening to your wife and kids, and therefore you missed the boat. It wouldn't hurt to admit these things to your wife and let her know that you know you missed the boat. Let her know you're willing to take another ride on the boat if you two can work it out, but you're also willing to let it go if it's just 'beating a dead horse' for lack of better words. You NEED to be flexible, you NEED to listen and hear what's being said, and you NEED to be honest with her without making her feel guilty or blamed. I know it sounds like a lot, but if you want her to talk to you and work things out, you need to be open to change... and not make her feel trapped (like you'll only talk if it's going to work out).
Secondly I would call your wife, set up a time to meet. Let her know that you want to make life the best it can be for the kids and you'd like to sit down with her and work out what that will look like for you and her... regardless of whether you stay together or not. The thing that should be on the forefront of your mind is that your kids need you and her to be civil and make the tough decisions so they don't have to deal with any more pain then they have to. You need to be the bigger man and take life's punches as they come.
There may be a chance to work it out with her, but there may not be. You need to be prepared to deal with either outcome. Once this has been decided you need to outline the steps you will take to 'get there'... to the place where things are working in a manner that the kids are in the best place possible for the given situation. You may need to take whatever crap job you can find, move to an apartment close to her so you can see them often, etc. etc.
You can also get on with your life and start making decisions that will put you in a better place once you know where you stand with her and the kids. Don't get me wrong, this is NOT easy. It's a long hard road, but the first step is admitting your mistakes and sucking up the consequences of them. From there you can rebuild your life and get on with finding happiness... a little older and a little wiser.
Sorry to hear of your losses and I understand how hard of a time this can be. I wish you all the best and hope that you and your wife can work something out and find a common ground with the kids if nothing else.