I very quickly found a YouTube video on this exact repair. It showed how to dismantle the mixer and confirm which gear had stripped. Once I opened it it was very obvious that the nylon gear that is designed to strip to prevent the expensive motor from being damaged was in fact stripped, so I ordered the replacement part. Once it arrived, I followed the video instructions to perform the replacement and the mixer was back up and running.
The repair isn't especially difficult, but it's a bit involved and fairly messy since there's a lot of disassembly to perform and the gear assembly is literally packed with enough grease to last a lifetime. If you're at all handy you can do it, just put down lots of newspaper first!