I'm at a bit of a loss as to what I can do. I have the same Kitchenaid Pro 600 mixer shown in this video:

I replaced several gears which were worn/chipped. All the replacement parts look correct to me. But when I put the housing on, the accessory drive gear has some play back and forth, such that it won't engage unless you push it in. If you let it go the motor will just spin without engaging the gear at all.

The gears that slide down over the planetary shaft fit perfectly, and the snap ring goes on in the right place. It doesn't seem like there's any way it could be done differently, so I'm really perplexed.

Has anyone encountered this issue or have any tips?