the first problem you are having is that you don't know what an illegal alien is.
So let's start there:
Out of Status vs. Illegal Immigrant
1) Out of status
Somebody who entered the U.S. "with inspection" which means with a visa or as part of the VWP. Such a person applied for a visa at the U.S. consulate in their home country. They had an interview and needed to submit a police certificate. They were issued a visa and then they were inspected again before being "admitted" to the U.S., at which time they received an I-94 (Entry/Exit) form in their passport.
The U.S. government knows everything about them; their name, their parents’ name, their birth day and birth place, their criminal history, work history, and even where they are staying in the U.S.
2) Illegal immigrant
Entered the U.S "without inspection" (EWI) by hopping a fence, swimming through the Rio Grande, or hiding in the trunk of a car. The U.S. government knows absolutely nothing about them. Not-a-thing. They could be mass murderers or serial child rapists, drug smugglers or worse.
The difference is easily explained by comparison:
You are having a party. You invited a guy to that party and when the party is over he's still staying. All you need to tell him is to leave, and if he does, you have no problem. That's the guy who overstayed.
But if you don't have a party, come home, and there's a guy who climbed into your house through the back window, and you have no clue if he's just a thief or worse, that's the illegal immigrant. Here you need to call the cops and they have every right to arrest him for a crime.
Your second problem is that we have millions of "real" illegal aliens in the United States, and since deporting a single one would cost the taxpayers $11,500 on average, we can't afford to deport more than 400,000 per year, all on credit, as the U.S. government is broke. Therefore we do not even deport illegal aliens if they haven't committed a crime. So in about 40 years, when all the illegal aliens have been deported, we can look into people who overstayed.
Now, back to your question: if you have prove the person uses a fake ID and SSN, then he's guilty of a crime, as document fraud is a felony. In this case take your proof to the police. If they investigate and he's being convicted and sentenced to prison, ICE will put an immigration hold on him so once he's being released, he'll be deported.
The son of a German mother and a Swedish father, I have lived in 6 European countries before immigrating to Southern California two decades ago. I work as a corporate attorney in Santa Barbara and answering questions here is my way of giving back.