Your next bet is to contact your own car ins co. first

They can give you much more accurate direction in this case. Call your ins first, fax them your police report, file a no-fault case, they will take you to the other insurance company. But I do not know what that ids insurance is that you mentioned before. I live in NYS and only dealt with Allstate(my ins) and Geico(the car owner's, not the driver's ins.). The at-fault party normally is easily determined by the case handler if the that ids IS the other driver(or the car owner's) ins co.
Good luck.