Have you tried those Mr. Clean Eraser Pads yet on the yellow stuff? I've had relatively good success with it, cleaning the white, padded plastic handles on my fridge as well as the outside of the box. I've also used them on the white painted kitchen cabinets and my 20+ year old Cuisinart food processor. Granted the food processor isn't back to new, but it did take it up a few shades lighter. Though you're just supposed to use water with the pads, I've also tried them with dish soap and Windex. They'll also easily tear, but they keep on working.
Another trick I've tried is very, very carefully using Brasso polish. Test the plastic first to if it may scratch or melt. I've been able to restore old paint on cars with it and the painted surface of an old Kitchenaid mixer my Mom got me at a garage sale many years ago. Brasso did bring up the finish on a couple of old clock radios I used to have. It took off years of cigarette smoke and polished them up very nicely.