If you think about the forces your wheel encounters as it rolls over the floor carrying 30+ pounds, being twisted all over the place, it might be unsurprising how the attempts so far with glue/paper have failed.
These things retain themselves by the hole in the plastic carrying some sort of restriction so the wheel can be hammered onto the shaft, and the restriction (perhaps a ring of metal with a split in it) opens up temporarily then closes around the thinner section of the metal shaft, so it's gripping the metal (loosely) but is also too big to let the wheel come off again. Ropey 2D cutaway diagram time:
- Black - wheel
- Silver - peg with thinner section
- BLue - metal ring with split so it can open up (it has enough space inside the wheel)
This blue bit has dropped to pieces, or been lost, or the plastic is destroyed and it isn't hidden inside any more. At worse, the wheel is really cheap and the blue thing wasnt a metal ring at all, it was just some plastic fingers that pushed out of the way when the wheel was hammered on and flexed back (and theyre now chewed up and gone):
I would consider 1 of 3 options:
Just replace it, if you can find a suitable replacement. Engineering a fix will take more time-is-money than the replacement will cost
Put it back in place, and drive a couple of screws through it and into the mounting block, holding it in place and stopping it from turning. Don't roll round so much while youre sat on the chair as it will wear the wheel out, but the other wheel will take some of the load, and it'll not wear out so fast if youre not wheeling it fully loaded. Don't put your desk on the chair and roll it across your mom's new travertine floor though!
Obtain a shanked bolt* that is the diameter of the pin that holds the wheel. The length of the shank should be equal or greater than the width of the castor as a whole (the distance between your fingertips if you touch one wheel center with the left index fingertip and the other wheel center with the right index fingertip). Drill a hole in the center of the broken wheel, drill a hole in the centre of the other wheel, drive the pin out of the wheel hub. Reassemble so the shanked bolt goes through the whole assembly, replacing the pin that carried the wheels. When the wheels roll, they should rotate on the bolt shank, not the bolt threads (the threads will act like a saw, chewing the plastic). Affix a locking nut to the bolt. Do not tighten the whole assembly to the point where the wheels cannot rotate
A shanked bolt looks like this, the smooth part being the shank:
The whole repair looks like this:
- Black - plastic
- Silver - bolt, white diagonal lines indicating threads
- Orange - locking nut