Here is THE long-term solution.
Announce that there will NOT be any more "nod and wink" illegal entrances into the US,and that anyone caught will be deported and FOREVER lose their right to become US citizens or even Resident Aliens.
Then,to prove we are serious,we start rounding up the ones that are already here,and deporting them and denying future citizenship or legal alien status to THEM,too. This will help the ones that really want to be here to self-deport before they are caught,and maybe inspire them to apply to come here legally.
NO MORE AUTOMATIC WELFARE ENROLLMENT. You either come here with a trade that is needed in the US and start working,or you are deported.
If the Catholic Church and others are so desperate for new members they are willing to pay for all your upkeep and living expenses until you get a job,let THEM pay your bills.
NO MORE CATCH AND RELEASE!
DEPORT,DEPORT,DEPORT!