If you can't see any chipped or missing teeth on the ring gear and/or pinion gear, just replace the spider gears. Not sure what you'll get into in regards to parts. Sometimes spider gears are only sold in kits. The kits include all the parts needed to replace all four gears, the pin and the thrust washers. Sadly, this will require removing both axle shafts. The gear with internal splines runs on the axle shaft, one for each side. The gear that's smooth on the inside spins on the pin, one for each side.
A trip to your local JD dealer will clarify all the parts questions. While there I would get a mechanic involved. He'll tell you all the other things to replace while there such as axle shaft seals, etc.
He can also offer any suggestions in regards to things you should do to minimize chance of future failure.