Sweetheart, I know this feeling. It spirals doesn't it and you feel you can't go anywhere without bashing into a wall.xxxxxxxx
First things first, your children need you and you need them. You are the only mother they will ever have and no matter how crap a job you think you are doing, the very fact you are on here being heartbroken is the very proof that you are an amazing mother.
One of the things that got me through depression (and its not totally gone, more like a dodgy friend I have to tolerate) was learning I could be loved left, right, up, down, ugly, back, front, shit days, good days. Even on the worst day something can be found to remind you that there is good in there somewhere.
Look for a photo of when your children were born. Remember how you felt then.......incapable? And how you learned. Frightened? And how you know those fears are not so big once you face them. Happy? Because you laboured and came through it with a special bundle who will learn all from you.
Talk to me about the through the night waking. It is the main and first thing to tackle. Without some time off at night you cannot begin to fix you. And you are fixable xxxx.
I have three children, two of whom are/were bad sleepers. I have seen a variety of problems and have overcome most of them. I can recommend a book I found hugely helpful. It might take more work in the beginning but I am sure we can get the through the night thing bettered with some hard work and commitment and support.
Whilst you are not getting decent sleep you must try to nap whenever possible. A body doesn't need consecutive hours of sleep to feel better, short bursts are not the best but are ok to maintain a body.
What age are your children? How is your relationship apart from a little lack of support through the night?
Before you can get anywhere, you need to feel empowered. Talk ....................
If I can save one person from feeling like a failure when they are pouring their heart and soul into their children then I must. You are just needing loved until you feel better.