If the contaminant was water-based, water will be the best thing to flush it out with. Unfortunately, "flush" implies a "through" motion and you are looking to clean out, among other things, the interiors of a bunch of tiny nearly-closed boxes.
Adding some sort of soap is great for getting rid of the dirt, but adds even more dramatically to the rinsing problem.
Getting all the residue rinsed away is, for all practical purposes, impossible.
To use a really bad pun, you are hosed.