the broken part showing in your pic is not teh shaft, again it's called a hand wheel adpater, thats singers name for that part.
Look at it, ( I see this in the photo), your looking for a pin, like a small nail. That has to be removed then the adapter will come off. your other one can go back on.
That pin. it's tapered, so one end is bigger than the other. it will only come out one direction. I use a nail set as a punch. mainly I have many wore out nailsets and those have a long shinny end. This is about the only part on a SM to do this, when you knock it loose, hit the punch / nail set hard as in a quick shocking blow, do not go tap tap tap.
In your case part of that adapter is broken away, that pin may also be bent and a Pain to get it out
It sounds like you have all or most of what you need. know this. The spoked wheel is heavier, when used with a motor. the machine will not stop as soon as you let off the pedal. you will get two more stitches before it stops.
I sew every night and this week am using a early 15, spoked wheel and motor. I like it, fact hate the small solid wheeels