Thanks everyone for your suggestions. I think I've found what I need:
Rug Doctor. Although I'm tempted to just buy a wet/dry vac since they seem to retail at about $150.
Maybe what I'll do is take the cushion covers off and wet vac the death out of them, and try the baking soda method on the couch itself (I reckon the cushions are most likely to be problematic if they get wet).
Now I just need to look up some pagan chants or bible verses that I can recite while I sprinkle this thing with baking soda.