The fix is this:
Make employers 100% responsible for any "Guest Worker" they want to hire. Make them pay their taxes, health care and welfare etc.. Make background checks required and give them an official ID.
Then crack down on the borders with the military. Have them conduct rat patrols and put a GPS chip in all the illegals they catch. Move the military troops from europe to provide the manpower for this. The GI's catch them then turn them over to the border patrol.
Think about it - this is exactly what we're doing in Iraq right now so it could be good training.
The hardest part will be stopping the cheaters and the companies that cheat. Make 20 years to life the punishment for the CEO's if that's what it takes !