Is there any way you could get a work visa or apply for residency being illegal?
The US sometimes declares amnesty programs for illegal
immigrants; it may be that you qualify for such an amnesty although
I would be in no position to determine that myself. Other than
that, the legal procedure would be to leave the country and then
apply at your local embassy or consulate for readmission. It would
then be up to them, to grant or withhold a visa.