I am fully in favor of the government deporting every illegal it happens to encounter. But I do not advocate mass sweeps, roundups and deportations and neither do most others who have given this problem serious thought. For all the reasons you stated, that solution will never fly. And why should Americans should be taxed for that cost which Chertoff has correctly estimated to be billions of dollars?
Instead, we need rigorous workplace enforcement and we need to cutoff all access of illegals to social services (except emergency medical care). When the jobs and the benefits dry-up, they will slowly self-deport the same way they came: on their own two feet and at their own expense. It took years for the problem to get this bad and it will take years to solve it. The key is to reverse the flow.
I agree with the approach of cutting off both access to the jobs and to the various social services. That will work fine until the courts issue some order that it's "unconstitutional". Hell, it'd be progress for me not have to tell my bank's ATM machine that I want instructions in English. Banks are federally-regulated so let's start there.