Exactly what coverages do you have? No collision = you being SOL. This is probably the case, which would be the reason why your insurance company won't pay.

Probably isn't even worth going after the other party. In FL you cannot go after their first home or car. You could end up garnishing wages, but knowing that the person didn't have insurance probably means they don't earn much money.

So basically you'd waste all the money you could have used to fix your car on a lawyer and you'll get shit in return and be no better off (or possibly worse) than you are right now.

Plus judging from the damage the accident will be ruled with the higher percentage of negligence going to you (front end damage in an intersection means you didn't yield to traffic)...so going to court won't even pay close to the total damages to your car.

As John said get a small personal loan to fix the car and just be happy you weren't hurt in the accident.