The pull chain controlling the fan speed snapped off inside the unit. How much disassembly is required to replace it?

You should not need to drop the fan for this repair. Remove the bottom cap, or light fixture it there is one. You'll see the switch inside attached with a thumb nut on the outside. Remove the nut, push the switch inside and it should be exposed enough to remove the wire nuts attaching it. Of course all of this should be done with the breaker and the switch off. The pull switch replacement is a coommon item in the ceiling fan isle at most home stores.

The hard part is getting the exact replacement switch, many look alike. I would contact Hampton Bay and have them send you one. It should be under warranty.

Ceiling fan pull chain replacement and repair - Ceiling Fans N More