Are you saying that Luxembourg is prosperous because of mass illegal immigration?
It is about legal and illegal immigration levels. The first sentence says reduce legal and end illegal, with the only question being, what is the "traditional and sustainable" level of legal immigration?
As to being prosperous and how immigration affects that, read Reason #9 in the article you posted. "Its true that immigration grows the economy, but so what."
The reality is that if the US had a realistic immigration policy, there would be but a few illegals here. By "realistic immigration policy", I mean at a rate higher than it is but lower than Luxembourg's. Somewhere closer to the immigration rate of Canada.
Luxembourg has a huge amount of legal immigration as it is one of the centres of the EU, so tons of money pours in from THAT front. It's also a top-notch banking location (think Switzerland, only more secretive) and you have highly qualified bankers coming to L'bourg. Those are the kinds of people needed in the US.