How long will the smell of a decaying corpse last?

It can linger for years if not cleaned properly. Many pet stores sell an odor reducing product or you can make your own with hydrogen peroxide,baking soda and dish soap.
This home made cleaner/odor reducer will "bleach out" some materials however so test it first.
The formula is one 500ml bottle of hydrogen peroxide (available at any drugstore/pharmacy) 1/4 cup baking soda and 1 tablespoon any type of dish soap.
Mix this in a wide deep bowl until the baking soda is dissolved and pour into a spray bottle. Apply solution to area the cat has damaged and allow to sit for 10 minutes, rinse with water and absorb with clean cloth, allow to dry and repeat once more if necessary.