I tried to go to the source link for that data, but it was dead. Not to mention that those children's identities were not stolen solely for this purpose of utilization of illegal immigrants, but of were instances of general identity theft. You and the author inappropriately correlate that. The same goes for the 7 million figure. In the New York Times article it cites, it only references W-2 forms in a general sense, not specifically to undocumented immigrants utilizing them fraudulently:
This whole piece is just a bunch of garbage. In its "A Culture of Corruption" section it tries to insinuate since these people come from societies with obvious levels of corruption that they are also inherently corrupted themselves. Now there is some racism so soft you could sleep on it. Undocumented immigrants in this country are less crime prone than the citizens that already reside here.
The Mythical Connection Between Immigrants and Crime - WSJ
Now I obviously don't approve of identity theft, but clearly the best answer would be to facilitate a way for these people to honestly contribute in the way they would honestly like to.