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.
Good luck.