First of all, I will presume that "they all leave" refers only to the illegals, and not the legal immigrants (it's unclear from your wording).
Second, what happens to the economy and the jobs done by illegals is that once the illegals are deported en masse as they should be right this moment then we can see if legal immigration levels need to be raised.
Problems solved.
Cheney is obviously refering to legal immigrants. We don't want those folks to leave, I am merely creating a scenario.
You didn't answer the question, though, so I'll try another approach.
We can safely assume that many illegals ARE working. What happens if they all leave? Do all those businesses simply collapse until we get it all worked out perfectly?
I have no idea how we would get illegals to leave short of hiking taxes, which we are all against, to have the manpower to investigate immigration violations and arrest and deport the illegals.
I don't see anyone providing valid answers on this, just slogans and easy, impossible answers.
My answer (and I suspect others') would be to tighten border security now, to try to stop the flow into the country. Then we would issue a national I.D.--something many here are against for various reasons--and attack this at the employers's level, making every employee on a payroll match with its legal ID number.
But I don't have any expertise on immigration reform, and frankly those who have easy answers on both sides don't seem to have any, either.
No one can seem to give an honest, informed, straightforward representation of what can be done to remove the illegals from this country and what this would entail in cost. We can sit here and say "Well they should just stop spending on X, Y or Z," but that's unrealistic, especially with our current congress, which is spending like drunken sailors. And as this article points out, thanks to our wonderful government of some years ago, anyone born here is a citizen and thus their parents have a toehold.
I'm sorry but your solution doesn't seem to solve anything. Simply saying "find and deport them all" is asserting what has to be done--that's not a solution, it's the end result, like saying "stop all the criminals from killing and raping". And the short-term hit to the economy would be massive in terms of the money to pay for that and the loss of workers.