At the time of accident you should have written down the at fault's insurance carrier name and number or address. If you did do this and this information is incorrect contact Florida's Department of Financial Services which should help you get the insurance company's number.

If for some reason they are unable to help you the police that showed up at the accident should have been taken and filed a report. Part of this report is both drivers' information, including their insurance information. Florida's no fault law should not stop the police from giving you a copy of the crash report, especially since the other party was cited.