It is no secret that members of the big-business lobby (the U.S. Chamber of Commerce, et al.) are big donors to Washington politicians because they want cheap labor. Thus the dilemma of recipient members of Congress to satisfy their big donors vs. satisfying their own constituents.
What about this idea? Allow the businesses that want the cheap labor to sponsor an illegal immigrant and his or her family. This would require the employer to provide housing, health care, medical care, English language education and food costs over and above what the low-income immigrant takes home in pay.
A U.S. Supreme Court ruling from 1982 (Plyler v. Doe) requires illegal children to be educated in K-12 public schools. Perhaps the corporate sponsors could make a donation to the local public school also?
Currently, American taxpayers are subsidizing all these costs.