The first thing is that on that long pin is where everything needs to pivot.  As I can remember, there is also a metal ring with a setscrew in there to hold onto the bottom of the mechanism.  I even remember having to repair this very mechanism on Daddy Sam, because the backspace was frozen into place on that long pin.  What I will do is to study the mechanics of it again and give a little more accurate description of how it all goes together.

Ah, thank you!

If it does what I think it does there's no way anyone could do it, and that doesn't seem right... 

I think the part you described is the sleeve that goes in with the rest of it.  I had a chance to look at both my 1949 Royal Arrow and a 1956 Royal Aristocrat.  The only difference is in the way the backspace bellcrank is held in.  The bellcrank is offset about an inch, with the bottom part facing the left side inside the typewriter, and the top part, facing the rear.  Both sides are hooked to linkages.  This bellcrank is held in place at the top with the 6mm (about a half inch) sleeve.  I'm again thinking that was the part you found.  The pin, or the small rod I'm talking about is about 2 and one-half inches long.  There are two ways this is held in place--you either have a solid ring with a setscrew (the old method), or a c-clip (new method).  If the rod has a groove in one side, you should have a c-clip.  If this part is missing, a hardware store should have one. 

To put it all together, you will need to remove the back of the typewriter--two bolts at the top in the back.  The offset bellcrank is probably hanging on the links.  If you have all the parts, or if you can get a hold of a junker somewhere, the long rod will go in through the top of the frame in the back.  Underneath that will go that 6mm sleeve.  Then the top of the bellcrank, with the top part facing the rear.  If you have that ring/setscrew assembly, that will go on next.  That gets tightened at the bottom of the top half of the bellcrank.  But, if there is a groove in the bottom of the rod, then a c-clip will need to be fitted down below the bottom of the bellcrank.  That should do it after the back is buttoned back on.  Let me know how you come out.