I like the idea of large, $10000 or more fines for each case of an illegal worker, in addition to what you stated. Make the risk of hiring illegals so high that it becomes extremely unattractive to anyone who would think of doing it.
Illegals come here and pay very little income tax, game the system. Then before they can be caught up with, leave the country and come back with another stolen identity.
I've seen estimates of 12 million illegals in the US. If the average illegal is skipping out on $2500 income tax not paid, that's $30 billion in lost revenue. Add whatever the cost for the medical care, education, welfare, etc. is and that's an extreme amount of money to ignore.