If you are not sure you had the green arrow (left turn trafic light) then 99.99% sure there was no signal and you jumped the light. Otherwise, it must be the other party that jumped a red light and hit your car if you have the left turn right, which should not be ignored in the first place.
I do not believe there will be 權威部門 that can do anything for a small fee. The fancy lawyer stuff you see in movies does not work in real life. Even if it can happen, you probably have to spend tons of more money than in a settlement.
Personally, I believe it is almost impossible to prove whether the other party was speeding or not.
It is too late and too expensive to do so.
The impact of the collision was not determined solely by speed but also angle, vehicle type, tires, and the surface of the road. So your camry 被撞出約60英尺 does not necessarily mean he was speeding. I saw a similar accident three years back. The two cars colided and none of them moved an inch but they were both driving at about 40 m/h.
Find a good lawyer and try to settle it.
Good Luck.