All that is needed to correct the problems of immigration is to make hiring any illegals a terrible choice for any employer.
Make it a mandatory jail time felony, first offense. Make it a heavy fine, so it cannot be a cost of doing business.
All these other ideas, 10 year wait, fines, background checks: all unnecessary. Go into a few states, and bankrupt 20 or 30 employers under the new law. The rest will get the message.
Now, the last thing the GOP wants is the flood of cheap labor to end, but that's the price.