What to do if I may be at fault for the accident but the other driver had no license or insurance and may be here illegally?

If the police were called to the scene of the accident, in many jurisdications, the police would have arrested the other driver for driving without a license.  In many jurisdications, once arrested, ICE would place a hold on the arrested illegal alien and commence deportation proceedings.  As far as your fault in the accident, that's a separate issue.  You could be sued for damages (property and bodily injury) if the Plaintiff's attorney can prove liability and damages.