Are you in Australia?
You can buy the entire unit for $132 AU dollar.
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Please look at the following picture and see are they the same control layout.
Since the motor is still working well. You can pop the "S" Sunbeam button out and un-screw the knob out for a quick inspection.
The plastic on the center (stem) screw is probably cracked.
When you turn the dial,it does not rotate the center (stem) control shaft.
Strong epoxy mix will bind the broken plastic knob back together.
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If all else fail then I will recommend an external electrical control unit call AC regulator unit.
It's use in-line between the AC outlet and your Sunbeam A12 mixer.
It's not a pretty unit but you can hide it under the counter top.
I will look up the price for you later.
I hope you can do a good cleaning and get the Sunbeam back in service.