How to determine why the whirlpool refrigerator has stopped running water

Here's an outside the box idea (which assumes that your main problem is that it wakes you up):

Get one of those timer plugs like they have for holiday lights and let it "unplug" your fridge while you sleep. You'll need a beefy one to handle the compressor spike when it kicks on (check the ratings on the timer plugs and compare it to the max rating of your fridge). Since the fridge door won't be opened, your food should be fine, though I'd test it the first night to be sure with a fridge thermometer. If your fridge is still cold in the morning, you might have a solution!

Are you on a city water system, or a well? Do you get the film from your tap? If not, then it's something in the water line to the fridge, or in the filter. In my whirlpool fridge, the "reservoir" for the chilled water is simply a coiled up roll of the plastic tubing. Clean out the fridge, and take a look at that to see if you can see any obvious discoloration or contamination - it should be clear. You could try replacing the tubing if you're suspicious.

If there nothing else obvious, then I'd try changing the filter, and run water through the new filter for a bit to clean everything out of the lines. The filter, after all, is designed to collect gunk from the water, and if it's been a while, it could simply be full.