I would suggest this policy:
1 - 60 day moratorium, if you self identify, and leave, you are still able to apply and get in line.
2 - At day 61, stop ALL benefits, let them get the benefits for 60 days to finance their exit.
3 - Day 61, if you are still here, you are caught deported, and lose the opportunity to every reapply forever. Personally, I would include all family members and relatives. DNA test every one caught, and test everyone asking to come in.
Although I do not have a fix, I am against a merit based system because if we are taking the cream of the crop, what are we doing to the countries they come from. Maybe the cream of the crop should stay there and try to fix their own country.