The answer is easy, but in practice it is very hard. Yes, I had my heart broken in a million pieces more than I care to admit. I also had young children to tend to. My answer is just that. Tend to your children. In tending them, you tend yourself. The pain is real, and feeling it is ok. But do not act on this pain in any destructive way for you or your children. If you or your children are in danger you must take appropriate actions to protect yourself.
Feeling heart broken is normal in the course of one's life. You will survive it as long as you act responsible.
Some depression is also normal from time to time as we course our way through life. Only you can determine if your depression is temporary or not. Again, if you are acting responsibe, then it probably is temporary.
I do hope you find the strength inside yourself to carry on in a responsible way. For your sake, and for the sake of your children.