I don't think a wall is as "insane" as a lot of people claim, but if you ask me, the way to go about this is to "sour the milk." Spend 4-8 years doing a federal audit of employment. Anyone found employing a person illegally in the U.S. gets a year in jail per illegal hire, plus a $1mil fine to the company each. Also, the jail time applies to every single manager from the hiring manager up - they all get the slammer. That will prevent boardroom suits from pressuring middle management to break the law, because it will be their ass, too.
Also, naturally, any state, county, or city functionary who tries to block enforcement will also get jailtime. No more "sanctuary cities."
When the jobs for illegals dry up, they'll self deport. They got themselves here, they can get themselves out.