I'm not sure exactly which speedtrap you are referring to, but try the following things:
1. Attempt to approach the speedtrap from the opposite direction. Sometimes this helps if coming from the opposite way is more of a straightaway. Or even if it isn't, there might be less traffic.
2. Wait until you have a faster car and then come back to the event. The CLK500 is fast, but there are faster ones.

3. Tweak the settings in the pause menu. The sliders could help you get the faster acceleration you need, at the cost of top speed, but you don't really need top speed, what you need is 111 mph in a certain distance.
4. This one really worked for me: Instead of slamming on the nitrous, pump it in short bursts (tap the button). It allows for better control (so you don't slam into traffic and walls) and for some reason I can't really explain, it'll get you to the speed that you need.
I hope this helps!
