Why wouldnt John have written a full letter to the Detective?
If john really wanted to bring the corrupt Lt. down I don't think he'd have just sent a single Polaroid.
since he was safely in Mexico there wasn't any reason not to tell his entire story.
Also letting the detective know the lt. had all the rest of the money would have been a useful tip. They could check his bank to see if a mysterious large amount of money suddenly appeared in his account.
Of course how did John know who the detective on the case even was? If he called the Police Department would they tell him that information? Couldn't there have been more than one detective in the L.A. Police Department working that case?