Here's a pic of the 2011 Wrangler ignition cylinder from the parts catalogue. Difficult to determine what;s going on exactly but it would appear the cylinder itself is removed from its housing.
Looks like two springs and it looks like a big thread on the cylinder, those two springs appear to correspond with two pins. Do you see anything there that can access those pins, two tiny holes maybe? Very difficult if you depress the two pins, maybe the cylinder unscrews from the housing. Does that silver plate guard remove where there might be access to two holes?
What are the instructions for the 2007 you found?
1. In order to break like that was the key bent thus jambing that broken piece inside? If you stuck a probe in there does that piece of the key move or is it wedged in solid?
2. Since you're going to get a new one anyway, you sure you don't want to try to stick a dab of crazy glue on a thin rod make contact and let it dry then try to pull it out. Assuming it's not wedged in there and moves if you probe it as above.
3. I'd definitely try the plasti dip, maybe you're not familiar with it, if you spray it in there, build it up with a few coats, let it dry thoroughly then peel around key entry and pull straight out, if that piece is is free moving, it may very well come out. Have a look at all the threads in this forum referring to plasti dip, its sprayable plastic and removes once dried like this:
4. Can you call your service dept. at the dealer explain the whole assemply is dismantled and can they or a mechanic simply tell you what the next step is to remove it.