I don't remember when I did it--it was after my sister left--but I dug the high-back chair out and lugged it upstairs, because I'd had a thought about how I could fix the broken caster.
I thought, "What if the casters on the low-back chair are the same kind as the ones on the high-back chair? Is that possible? I should check it." Regardless, I wanted the high-back chair, so off I went.
As it turned out, both chairs used the exact same kind of casters. What are the odds?
...considering the likelihood that inexpensive home office chairs come from the same manufacturer, or at least that the various manufacturers probably buy their low-cost casters from the same source?
So I swapped the bad caster on the comfy chair for one from the non-comfy chair, and proceeded to enjoy sitting at my desk again. I've got the comfy chair back; I can lean back and relax and play WoW (or whatever) and not have to sit upright.
"Why," you ask, "did you switch in the first place?"
Two reasons: one caster was broken; and my lower back was hurting all the damn time. I thought it might have something to do with the chair (and it got better once I switched chairs).
Anyway, it's all good.