Here are couple of mistakes that you have made.
1. Failure to call the police at the scene. If you had called the police they would have made a police report and issue a citation (ticket) for the lady.
2. Called her insurance provider then and there and file a claim and at that very moment the insurance agent would have asked the lady if it was her mistake or not.
Solutions:
1. Best thing for you to do right now is that you have her plate number so go to the nearest police station. And ask the officer that you want to file a police report. And tell your situation that you forgot to jot down her insurance provider.
If you dont do anything here are the things that could happen.
1. The lady might get away with this mess and get her car fixed and later quote that no accident ever happened. Even if you were able to find out her provider later.
2.She might have your insurance policy number and provider (if you both exchanged the information). Now she might claim that you were the one who hit her. Since you failed to do the first (1) and (2). Because she knows that you dont have any witness.
3. Both of you keep quite and you would have to pay to fix your own car. And she would fix hers. End of the story.
There are other things that could happen as well if she wants to be nasty with you. I guess every mistake makes a learn. I wish you good luck.
Passionate Dreamer,
YekloYatri