I am sure that I will get flamed for the heresy that I am about to profer, but...
There are a lot of people that have problems with SB600 that either won't startup at all or won't fire.
1) You have been given several hints for improving battery contacts. One more would be to use a paper clip or some such to pry up the contacts on the bottom of the battery chamber. 2) Sometimes the battery chamber can get jostled & it will hit and dislodge the inductor below it from the printed circuit board. Are you comfortable with a soldering iron. WARNING! There is a charge built up on a condenser that if discharged could stop your heart. Here is a link should you want to chance it. http://www.ifixit.com/Guide/Disassembling-Nikon-SB-600-Speedlight-Bottom-half-of-the-flash/7682/1 3) The other thing that can go wrong is breaking of the bulb (fresnel?). Some have found the part & repaired it themselves. 4) Some have reported repair charges from Nikon at about $130-$150. 5) I think you have to weigh the risk reward of trying to repair it yourself or having someone else to repair it or purchasing a used working one or a new SB700 (or other) with warranty.
Try googling or binging "SB600 Won't startup", etc. to see more information.