For a broken shower knob, I wrote an instructable showing how to print an emergency replacement.
https://www.instructables.com/id/Printed-Shower-Knob/ In the comments, Studleylee indicated a method he used to repair such a handle (hopefully he will write his method up). This triggered an idea for repairing my original broken knob.
All I really needed was a way to hold the broken piece to the original part of the knob--a big fat washer would do the trick.
I made a few measurements with a caliper (these are available on amazon.com for less than $15.)
I make a quick pencil sketch of the actual measurements and (circled) the size I want to create.
This takes less than one minute to design in Autodesk's 123D software (free).
In "Primitives," you create a cylinder. The cylinder has a radius of 14mm and a height of 12mm.
"Draw" a circle in the center, radius 7.53mm.
"Extrude, Cut" the 7.53mm circle for 12mm.
Print it--takes 11 minutes and uses less than ten cents worth of plastic.
Take this special washer and friction fit it around the broken and intact pieces of the knob.
Screw this in place and you are done!