My problem was a little different than either of theirs. I had repaired the connector in the past -- the cord frayed and shorted right near the DC connector (the part that goes into the computer) so I spliced it and tried making a better strain relief. That lasted for a couple years but it broke again so I figured I'd try and replace the connector entirely.
Step 1: Disassemble the Old Connector
I used hot-glue as a strain relief in my last repair so I used a heat gun to get the glue off. I don't remember how the connector came apart originally, but the gist was to cut off the plastic shell then pry apart the metal shell. These are the pieces I came up with.
Step 2: Selecting the Replacement Connector
I decided to hedge a bet and just install the replacement jack without the original metal outer shell.
Although the outer shell was connected to the middle-ring and to the wire shield, I guessed that the shell itself was not connected for supplying power. I would have to take apart the iBook to find out for sure, but for the time being I just assumed it wasn't connected. If the resulting repair failed to get the iBook powered, then I could assume I was mistaken.
Step 3: Prepare the Wire
First, remove about a half-inch of outer insulation without damaging the wire braid. I find that if I use a sharp blade with light pressure against the wire, I can gently roll it along the blade and it will cut nicely. If it isn't obvious, it's easy to cut your thumb, so take your time and be careful.
Next, to separate the braided insulation, I use a pick to comb out the wires. Some electronic shops sell them as dental picks -- it's a handy tool to have. Once the wires are fanned out, I pull them to one side and twist them together.
Usually I'd leave a long tail on the twisted braid, but since I'd be connecting it "backward" (the braid connects to the inner-ring and the center-conductor connects to the shell/ground) I cut it short and tinned the ends.
Step 4: Solder on the New Connector
Step 5: The Moment of Truth
Once everything was done, it was time to try plugging it in. As indicated by the orange glow, it appears to have worked!