Hmm... decades.
The atmosphere isn't very conductive, so the current flow into even a very large metallic structure is very small (microamperes). The average 9V battery has 200mAh or so capacity, so will be depleted by 1uA in around 22.8 years.
Yes the atmosphere is charged, but that doesn't mean you can draw power from it. It's like saying, yes, a spring holds a force, but you can't just look at it to get power out of it. You have to cut in and complete the circuit to discharge it.
That said, if a space elevator is ever built (of whichever type: centripedal belt, "beanstalk", or what have you), and it penetrates the ionosphere, it has the capability of interacting with this charge, because the ionosphere is appreciably conductive. Billions of volts would be available along its length, at a pretty fair current I'd imagine (amps?), at least until the ionosphere is discharged to equilibrium. This would probably alter global weather patterns! I don't know if anyone has considered or analyzed this effect in a space elevator proposal.
Tim