Still, you should be able to fix this. For starters, I'd fill a bathtub with hot water, throw in a cup of laundry detergent, and soak the thing for a day. Swish it around a few times, and to the extent that you can, wring it out once or twice and re-soak it. Then take it outside, hose it down thoroughly to remove the soap residue, and air-dry. I'd be very surprised if that doesn't work.
But, if it doesn't, stronger measures may be required. Repeat the bathtub procedure, but this time throw in a cup or three of Lysol All-Purpose Cleaner. Let it soak again. The Lysol should kill any bacteria hiding inside the foam backpad. You can also spray it with Febreeze, a product that I've found is very effective at completely eliminating odors, rather than simply masking them.
If none of those measures works, all I can say is hold your nose!