Nice try, but elasticity isn't the issue, since the last time anyone looked in America there aren't assigned jobs and the labor force is free to move to where the money is.
Now, let me explain it in simpler terms so that you will have better understanding: Turn your page to "market clearing wages". Bid up wages and surprise, surprise, you get more people willing to do the work.
But then, those who have been making money by using illegals don't want to pay market wages. They prefer to insist that they deserve to have their labor force subsidized by the taxpaying public. Somehow they shouldn't be subject to the economic forces everyone else is, such as going out of business if they aren't competitive.
Your problem ie that you try to make illegals something extradinary. They are simply more imported labor. Something to take up the slack while Congress hashes out the conflict over permenant versus temporary.
BTW, your phrase, "don't want to pay market wages, indicates that you are mixed up. The wages that illegals are paid, are market driven. OTOH, wages to guest workers are set by the govt, a prevailing wage. This is why so many H2A and H2B guest workers go awol, they can make more money as an illegal.