It is very easy to get rid of them using no new laws and being no more intrusive than the government is now. Citizens have to present a birth certificate and proof of residency to put their kids in government schools. Criminal aliens don't. Enforce the law and when a criminal alien brings his kid for the "free" education arrest them both and deport them.
When they catch an employer working criminal aliens use the RICO laws to confiscate the business and anything the owner bought with his illegal profits. If it is a homeowner employing a criminal alien to do yard work or house work, take their house under RICO.
They should start the program by confiscating Tyson Foods. Under RICO they don't have to prove anything the company has to prove they are not guilty. Take down Tyson Foods and maybe confiscate a few homes and nobody would be willing to hire the criminals.
Demand proof of citizenship for any welfare or taxpayer funded service and that will get the rest.
No new laws no stopping people to check their papers. Just verification of employment records and positive ID for welfare and the problem is solved. It wouldn't even take an act of Congress. The laws are already on the books.
If it is a homeowner employing a criminal alien to do yard work or house work, take their house under RICO
That wipes out a lot of political appointee nominees right there. Oh wait, would that be a bad thing? ;-)