I've never met a KA-601 in person, but I took a look at the schematic and the headphone output wiring seems utterly conventional. The headphone jack is fed through 220 ohm resistors from the output side of the protection relay. That's the same source as the speaker selector switch, so if the speakers have sound, then there has to be signal getting to the resistors. It would be extremely unusual for both resistors to fail.
If the ground connection to the headphone jack went bad, there would be very little sound in the phones, but there would be some. The difference between the left and right channel would be heard in the phones. Are you sure that there is no sound at all?
I think the most likely explanation would be a rather extreme case of oxidization (or other contamination) of the contacts that are supposed to make contact with the headphone plug. Here again, it's very unusual for that sort of thing to affect both L and R channel equally. But if you've got Deoxit or similar, you could try wetting one of your headphone plugs with the Deoxit, then plugging in to the amp, and rotating the plug back and forth a bit to get some contact wiping action.
Good luck,
chazix