It's not up to the DHS to adjudicate, that's the courts and ALL accused have the legal right to have their case heard before a jury, basic Constitutional Right for all persons accused within the US. Most of the time they will plea guilty and can be deported but for the ones who know the system they play it to the hilt.
I know people who have been deported. It's quite a simple process. The illegal alien is shipped back to Mexico or Latin American very quickly.
Illegal aliens are not dummy. They know that lying to federal agent is a felony, risking lifetime ban from the US.
Furthermore, if an alien waits for a deportation order from a court, he will lose the right to return to the US for at least 10 years.
Therefore, most illegal aliens sign wavers to accelerate the deportation, and then they are back in less than two weeks, usually working at the same job.
As I say, DHS chief Chertoff has become a parody of himself.