How about finding out from these illegal alien kids who exactly their parents are, then bring their parents to the border holding facility where the kids are located, then charge them with abandonment, neglect, willful negligence of their parenting duties, and child endangerment?
Then immediately deport all their criminal family right back across the border.
That will take care of a bloody lot of costs indeed, both short-term and for the future.
Or just start a rumor that the US is deporting all these children to China so they can be processed into pet food