Unfortunately, they would never be able to pay for their jailing. Not the way our penal system works. Last I heard, it cost something like $20K per year per prisoner. No way is their unskilled labor worth that.
But to say it would cost too much to find them and deport them is just nonsense. If there are 10 million of them, then that is probably 3 million families and you have to figure at least 4 million kids going to public schools. That's 40 billion dollars every year all by itself. Not to mention the cost of free ER medical care and all the other programs. Plus the cost to the economy of all these bozos driving around without insurance and speeding away from hit 'n' run accidents.
I agree the place to start is with the employers. Make it too expensive to hire illegals and illegals will stop coming and go home on their own. Without work they won't stay, so that is where to hit the problem.
With employers, you'd have to make it easy to have a positive ID or you couldn't hold them liable. A first step would be a database check to see if the SS number is in use somewhere else. If so, the employer would need to contact the owner of the SS number THROUGH the other employer to verify they are really the same person applying for the new job.
Even better would be to REQUIRE a valid State Drivers License or offical ID card with biometrics. Swipe the magnetic card, take a fingerprint, check both against the photo in the DMV database. Personally, that level of identity checking wouldn't bother me. In fact, I think every cop should be required to check an ID in real-time the same way any time they stop anyone. And be required by Federal law to hold that person for deportation.
The saddest part is that the people have a will to control this. It is only politicians and courts who don't for some reason. Californians overwhelmingly passed Prop 187 which was relatively mild but would have denied at least some social services to illegals. 9th Circuit shot it down.