I don't have an EF-S 10-22mm lens to play around with, but I do have an EF 17-40mm that is more or less the FF equivalent lens. Looking at the block diagrams of both at the Canon Camera Museum, they're very similar in terms of optical formula. They were also introduced only about one year apart.
To get to 17mm, the widest angle of view, the EF 17-40mm moves the rear group all the way to the back of the lens. I would expect the rear group of the EF-S 10-22mm to also move all the way to the back of the lens for 10mm.
It may be that something is out of place between the rear element and the front of the PCB in the back interior of the lens. But more likely is that one of the plastic/nylon rollers that follow the grooves in the zoom helicoids has become deformed or come partly apart and his getting stuck at 12mm.
Since you've indicated that you've been inside lenses before, you might want to take a look inside the back of the lens. Please be warned:
- This is risky, and could ruin the lens if done improperly!
- If things go wrong, you might change a lens that is usable from 12-22mm to a lens that is not usable at any focal lengths. Things will sometimes go wrong.
- If you give up and decide to send it to a repair shop disassembled, they'll almost certainly refuse to work on it. If they do agree to look at it, they'll charge (a lot) more than if you sent it in without being taken apart.
- If you break/strip/tear something inside and then put it back together as best you can before sending it to a repair shop, they'll be able to tell when they get inside the lens and will almost certainly refuse to work on it. If you want them to send it back to you they will likely bill you for the work they did opening it up and putting it back together the way they found it. If they do agree to continue, they'll charge (a lot) more that if you hadn't taken it apart, broke something, and then reassembled it incorrectly.
This YouTube video shows a disassembly and reassembly of an EF-S 10-22mm lens. However, I would strongly advise you to remove the two screws holding the communication contacts with the camera in place on the inside of the flange ring before loosening the screws that hold the flange ring to the rear of the lens.
Be very careful when disconnecting the flex cables from the PCB! If you damage any one of them the lens will essentially be toast. Mark the alignment of everything you remove before you remove that part. I'd also remove the rubber focus ring cover before removing the rear barrel and then wait until I had the rear barrel back on the lens to replace the rubber focusing ring. Removing and installing it with the rear barrel removed risks damaging one of the flex cables as you pull it past them.