"He got behind on his child support and got sent to jail for it THIS TIME."
So this isn't the first time he has not provided for his children.
"I don't know how throwing a guy in jail is supposed to enable him to pay his child support faster."
Was he on the brink of paying his child support the day he was 'caught' for non payment?
Does your brother have a job?
I dunno, maybe it would be more worthwhile to ask your brother 'while in jail' to start making a list of what he needs to do in order to bring about change in his life, so he can begin living as a responsible human being. With that list, examples of what he needs to do to make that happen. If he has trouble executing the exercise, are there not some resources in the jail, ie. a mental health counsellor to help him with that. I say 'more worthwhile' then just getting him out of jail to continue his life as he has always lived it.
Maybe it would be a show of respect to the family member who ultimately makes the decision to leave him there or get him out, to not to question her decision, if she decides to not do anything.