I would like to see meaningful data on how both "The border was fairly tight for most of US history" and how "Ike" cleaned things up so efficiently.

You may be able to solve your fence problem by having extremely strict laws punishing U.S. citizens that engage in employing illegals. I doubt any other approach would work.

I also don't know what the economic impact of losing the gray/black market labor would be and how long it would take to recover from it the changes.