Interesting and timely topic, I'l weigh in with some of my thoughts.

First, I don't think the issue of "they took our jobs" has played out, except superficially. And, I know there are more jobs now than even a year ago, and there are more people working now as well... but, the Bureau of Labor and Statistics employment numbers are grossly misleading as reported by the MSM. Even the BLS knows and offers this; Table A-15. Alternative measures of labor underutilization That BLS table shows an unemployment rate of almost double what is hyped by our Leaders and the MSM...

I'm sure we've all heard the point made "that illegal immigrants are doing the jobs most Americans don't want." I don't believe that is as true as most would like us to believe. When I was a young man, a lot of my friends went into construction building houses. Now, if I drive by a building site I see a lot fewer young white men working those jobs. I don't know what the statistics are, but just by observation I doubt I'm wrong, and I'm sure many young white men would like those jobs back. But, when Hispanics are willing to do the same job for less money, it's easy to see who is going to get those jobs.

You ask what the penalties are for the employer that hires illegal immigrant workers, this link may help, DOJ increasing penalty for hiring illegal immigrants

According to my link, two years ago the penalties were raised,

"the minimum penalty imposed by the DOJ for the unlawful employment of immigrants would rise from $375 to $539, while the maximum fine would go from $3,200 to $4,313."

and further down in that article,

"Violators facing multiple charges also would be subject to a new maximum penalty of $21,563 for hiring illegal immigrants."

So, as you asked for solutions, barring any courageous attempt by our Leadership to address Immigration, the only other alternative I see is to fine employers $100,000 for every illegal worker hired by the employer, with no cap or limit. And, I further propose that Civil Servants need to be fined individually $10,000 for each illegal immigrant they offer financial aid to. The idea is, if there are no jobs, and no financial aid available, a vast majority of illegal immigrants would choose to self-deport.

Of course the argument can be made that there needs to be a better system for verifying, who is a legal immigrant and who is illegal. In fact I'm sure arguments can be made on many aspects of my idea... My guess is if that plan was put in to effect, maybe by Executive Order, than there would be a whole lot of solutions offered by the business community and political leadership.

Personally, I think the Immigration issue is left unresolved by our Leaders on purpose. If Americans are in fear for their jobs, and safety, then they are far more likely to focus on the illegal immigrant as a problem in their lives, rather than the Leadership that is the cause of the problem in the first place.