Here's an idea : The Pentagon can offer a program where an aspiring young man (or woman) from Mexico or elsewhere can earn US citizenship by serving for 6 years in the US Military, provided their service is with honor. This is essentially a new class of visa, just like an H-1B, but is open to anyone who is young enough, able bodied enough, and determined enough to make this sacrifice to earn a US passport.
The military will have to start with a small intake for this program, and eventually build up the infrastucture and processes to take in a quota of around 200,000 a year. The US will have to organize bi-lingual officers (of which there are many) to manage these divisions of troops at first, and also organize some classes in which to teach them basic English as part of their basic training, but this is not too difficult. After these classes during basic training, however, these troops should be mixed among US born troops. Those who violate the miltary code of conduct should be discharged, and deported.
This is attractive for the illegal immigrants, because :
1) Being an enlisted soldier in the US military carries a salary of around $20,000 a year, plus several benefits that continue after military service ends. It is a more attractive package than they are currently getting by doing unskilled manual labor in the US.
2) When they emerge as US citizens after their 6 years of service, they will know a fair amount of English, and will be able to get much better jobs in the US, than they could have gotten before their service. They could even go to college.
3) The soldiers that do well may choose to continue their military career path, and rise up to higher officer ranks over time.
This is very attractive for the United States, because :
1) It will be a way to separate out which illegals from Mexico truly want to become Americans and contribute positively to American society. This will be a structure through which they can learn English and assimilate in American society with honor. This may even lead to some of them going to college, and contributing to the US economy at a higher level.
2) It will be a way for the US to boost troop levels, in case a major military operation is upon the US in the future.
3) It will irritate the corrupt elite in Mexico, as the people who otherwise were a burden on the public services in the US will now be strengthening America, and may even gain the political clout to influence change in Mexico in the more distant future.
4) It will infuriate military-hating anti-Americans who deride this as a scheme to cultivate cannon fodder, even as Mexicans who participate will have done so by choice. The same anti-Americans who were previously advocates of unrestricted illegal immigration will now oppose this, and will alienate them from Mexicans who were previously easily manipulated by these extremists.
This program is far from perfect, and many more details have to be worked out. But among available options, this could be a very attractive one. At the very least, there should be a program where people who want to risk their lives for America can earn US citizenship through a clearly defined path. I suspect the number of people who sign up would be large.