You didn't indicate whether the shaft slot is worn or the gear casting is worn. What's the condition of the old key ? Either way, shim it up with a new fat key and reassemble it. If the key breaks because of the lash, you'll get continuous miss-ties and the wiper arm won't clean off the billhook completely. That could lead to a few other problems, but no needle breaking if that's what you were worried about.
You could try to drill the gear and shaft in place and drive a roll pin to lock it, but the amount of slop you're describing is nothing compared to others I've seen. While you have it apart, examine the wiper arm ball and fix it if its flat spotted. Fill and grind it round and make sure it spins. If this gets worn, you will also have problems with knot completion.
BTW, mine has about 1/4" of movement (measured at the periphery of the gear) and I'm not a bit worried (yet).