It is a much easier task to address the influx. Most of it comes along a long, but defined border and through ports. Once in the country you have a very large space to monitor.
I taught in a school with a large number of students who came here illegally or who's parents came here illegally. The vast majority of them work off-book for family members or friends in small (2-5 people) businesses getting paid cash. They have fake SSN that are all but impossible to detect. The coyotes sell them as part of the package. It only got worse when the irs decided all kids needed SSN to stop tax cheats (law of unintended consequences). Those kids SSN are used for years undetected.
Part of my banking job was managing a fraud/risk unit. We 'caught' these many times when those now adult kids tried to get credit and had horrible credit. The sad thing is that some of those illegals naively believe they were given legit papers. The ones that get caught are an entirely different class of illegal...those who come on student or other legal means and then don't return.
The press would have you believe most of these people are working in processing plants. That is because INS used to target those companies because they could make headlines...very convenient during those appropriations discussions. The thing is, despite 'big' numbers compared to the 2 or 3 at a time, those places are not where most of them work. In many (not all) of those big cases, the company was being duped. Don't confuse headlines with facts.