in my own experience, if she says "no", its "no". I've never been cheated by any GF I've ever had, but if what you say is true, if she's lost the feelings for you (and if she cheats on you repeatedly, that's a big hint she has) it's all over. For good.
Dunno about kids, I have none. But I know about how destructive can be the life of someone who tries to win a lost cause. Or someone who doesn't know when the only option is to let the other person go.
Went through it for two years and a half a (long) time ago, for a girl who left me and I couldn't accept the fact it was over. No cheating involved, she simply did not love me anymore.
I did my best to win her back for the next 30 months. I shed gallons of tears, forgot I had a life on my own, was distracted from my studies, got some 40lbs of extra fat on me, stopped playing sports, stopped going out, lost all my friends (They insisted on me forgetting her and I simply didn't want to hear that chore anymore...I told them to f... off and well, they did. Thankfully they were comprensive some time later

) and all the time I was not with her, I simply was closed in my own house, wondering what I was doing wrong and what could I do to make her return with me.
In my case there was no fault involved by any of the two sides. Sometimes it just happens that whatever feeling got you together is lost for either the guy or the girl. She didn't cheat on me, she did care for me a lot (we're great friends up to today) and she didn't want me to suffer. But she also didn't want to go on with a relationship that she simply felt would not work as she didn't love me anymore. Only when I realized that I could do nothing to win her back (when she started dating other guy) and accepted I had to let her go, I moved on. And it turned out it wasn't that hard to do. Took me months (honestly, years) to get completely over it, but I recovered my friends, lost my extra weight, finished my studies and dated new girls. Enjoyed all of it. A lot. And realized that no matter how good she was (And she is a great girl) all the time I spent giving my best to recover her was not worth it. Noone is worth the hell I lived in for that long.
So my advice: let her go. Probably that means a divorce (should mean it but it won't be the first time a married couple doesn't divorce because of kids, but reach a "friendly" agreement between them). Do your best to be with your kids and try to settle a "peaceful" agreement with her that gives you good access to them. Regarding kids I won't say anymore, I don't have any and no experience with them, but people here have given very good advice.
In the end...whatever anyone says, do what you find is best for you and your family. But if you want her back be ware that probably it's already a lost cause. And the fact that you still want to be with her plays against you, The fact that you are ready to forgive her cheating says a lot about you, really, but unfortunately she probably won't feel it that way. Women have some strange instinct that keeps them far away from men they perceive as weak, and forgiving her cheating may be perceived by her not as forgiveness, but as weakness, getting you even farther from her and finishing any (slim) chance you might still have.
But please, if you see it's not going to work, don't insist, don't wait and don't stay for months like I did. Don't lose your mind as I did. In the end you won't have her back anyway and you'll regret all the time you lost (I do)
best wishes, mate.