The solution is to have stopped payments 10 kids ago. If a woman is on welfare, the agency should help go after fathers, assuming they are not in prison. But no more services or pay for more kids after going on the dole.
For the most part, the fathers of welfare kids are unlikely to have jobs whose wages could be garnished.
However much some people might want go after the dads, the only thing that will WORK is to make the mom's life as unpleasant as possible.