What? She walked in and gave away the most valuable thing in her life, in front of a judge.This is good news buddy. You get the kids, and your negotiations for spousal support just got way easier. This is very good news.
This right here is why you want a square girl for your wife. Don't put a ring on a party chick.
What state/country are you in? In the US there is no concept of "refusing to sign the divorce papers". You just have to serve them, something the local sheriff can do. And if they can't be found, you publish ads in the paper or something. Zero requirement for their cooperation in scheduling a date or concluding a divorce.
If one spouse refuses to show up to that scheduled court date boom you win whatever you put in front of the judge by default as long as its within the state guidelines.
I would give up anything before my kids including any amount of money.
Tell the judge the entire story and then say "if agree to pay $50 can I get a restraining order for both myself and my child against her?"