I could not find info on this unit so am not sure if the below is applicable.

Get some fiber tape or any type of tape that will not stretch lengthwise.
Wrap it around the knob shaft/tube a couple times as tight as you can to keep the crack closed.

Then get a small zip tie and tighten that around the shaft at the cracked area.

Also am not sure this will work as you may not have enough clearance.

Another possibility is take the knob to a decent sized parts supplier, they may be able to match it up with a knob from another unit. All will depend on how good/knowledgeable/nice the counter person you is.

It is the plastic knob that has a tiny crack - not the shaft that extends from the control panel.
Yes, I understood you were referring to the knob and not the timer.
I do not know if what I was referring to even has a name but it is the tube which is part of the knob and fits over the timer shaft.

I looked at the diagram for the timer and see that there is a screw that can be purchased separately. Is this the screw that the knob would fasten onto?
I do not think so. If you click on the picture of the timer, a new page will come up with more views of it. Click on these pictures and it will expand. The timer shaft with the screw end is part of the timer itself.

The screw is item #22 and the timer is #4 of the available parts listed for model KAWE550V.
Note that 22 comes up as the knob, so I assume you meant 25 which looks like it just holds the timer to the mounting plate.

You are welcome.

I do not know about the expert part, an real expert would have had a solution for you.