Even funnier, I just googled 'SB600 L301' and there are lots of people in the world with exactly the same problem. Some say Nikon placed the battery holder too near to this inductor and when the flash falls, they crash and the solder breaks. Sometimes the inductor's plastic case breaks too.

It's funny, but I have the same problem with the same part (the "black box", I think it's an inductor). I just disassembled the flash, so I haven't soldered the inductor back to its place yet, but I'm sure that will bring the SB600 back to life :) Thanks for the idea, I would have never checked the soldering of that inductor without this post, because it's on the proper place, just a bit loose when poking it with a screwdriver.