Bill Galston of Brookings and Bill Kristol of the Weekly Standard formed an outfit called The New Center to seek bipartisan approaches to public policy issues. They have published an interesting study on Immigration which you can find at
It is easy to read, or at least skim, and offers some basics about our current law, American opinions, and current legal and illegal immigration. It goes on to propose new policies. I don't agree with everything they have to say and might prioritize some policies more than they do, but the point of their work is that informed compromise is what it will take to solve this problem as opposed to constantly kicking it down the road.
I am not seeking to even have a discussion here about the merits of their proposals and certainly don't want to discuss who is to blame. I simply offer it to a bunch of smart people who I know are always looking to be better informed.