Report the matter to your own insurance carrier again -- in writing -- and give them all of the identifying information you gathered. Let your company find the driver and hopefully find their carrier.
In no way am I offering you legal advice, and in no way has my comment created an attorney-client relationship. You are not to rely upon my note above in any way, but insted need to sit down with counsel and share all relevant facts before receiving fully-informed legal advice. If you want to be completely sure of your rights, you must sit down with an experienced criminal defense attorney to be fully aware of your rights.
I would go to a precinct in person, explain your plight and ask them to run the plate for you. If that doesn't work, call your insurance carrier again and ask for the same. If you still haven't gotten anywhere call around to local attorneys, who may have access to driver insurance info. It may take a few calls before you find someone willing to help but be persistent.
Also - in NY I believe you can go on th Dept. of Motor Vehicles website to seek information. Perhaps you will find some info there. Good luck.
The aforementioned opinion does not constitute legal advice and is for general informational purposes only. See an attorney licensed in your jurisdiction for competent legal advice. No attorney-client relationship has been formed through the within legal question and answer session.