Children should be given equal opportunities in life regardless of their parents' decisions. Though the mother is clearly an idiot for saying that "someone should be held accountable" (you mean yourself? idiot), the children should not have to suffer for her lack of responsibility. If nobody is willing to support them voluntarily, then I must argue that this is one of the few cases where the coercion of the state is necessary and justified, not to guarantee equality of outcome, but to guarantee equality of opportunity for these kids.
However, the mother should not get a single cent for her irresponsible actions.
Somebody has to care for them to want to enslave people to have these children cared for. Otherwise, they would be left to die. The fact that it's an issue shows they will be cared for. It's only a matter of expressing that compassion morally without tyranny. In the end, if violence has to occur, it's only due to laziness from the people who send the brutes to get the children cared for. It's uncalled for. If you "care" so much about the children, go care for them yourself but don't force anybody else to do it.
In the end, it's just a desire: you want the children cared for. You may also want your lawn mowed and a steak; that doesn't justify violence to achieve these ends.