At @vaviurka's request, I got a nice new diamond dust sprinkled sharpening block from China and now we have five sharp knives in the kitchen (and three that were beyond rescuing).
It seems the key difference between an expensive block and a cheap block is that it doesn't come with the, uh, blocky part. So I made one real quick (ideal router job) and now it's a breeze to use instead of sliding all over the place.