It's a pretty easy fix. I'd buy a new joystick and replace it yourself though rather than repair the unit. It's possible but I've never taken that assembly to bits. The Tritons are built to be repaired so it's pretty easy to replace the joystick assembly. If it's like the Triton Studio (what I have) or Pro which I suspect it is then it's a case of undoing the large base screws, taking the base off and the joystick assembly can be removed easily with a few screws around the edges. There will probably be a long cable that attaches it straight to the motherboard and that's it. Careful with the cable when putting it back as they can get trapped and sheared if you don't route them properly.
You can get the parts from ebay usually but you might have to wait a bit for one to appear. Either that or Korg can send you parts out from their service center. They're pretty good, reasonably priced and have bits going back to the 1980s in stock!
Overview here on how to disassemble (should be the same): http://www.bustedgear.com/repair_Korg_triton-pro_disassembly...
Stripped it down and sold the keybed and chassis parts on ebay and made enough to buy a Korg Triton so I was happy.