White vinegar is a great household cleaning agent but it has to be thinned. (It's also a great weed killer.)
I guess I'll just wash the attachments and scrub off the oil varnish with a very fine steel wool. Odors dissipate over time and it's not too bad.
By the way, I've learned that the corpse the cat is leaving on your doorstep isn't a gift, it's a warning.