In my case, that something sticky was a whole ton of glue holding the front and back of the each ear piece together. There were what looked to be 4 blobs when in reality the entire ring was glued down. This meant that to disassemble the headphone I had to break the plastic ring in order to cut away the glue at the inside. Not pretty, but I eventually had the two halves separated.
Rewiring had its own issues. The original wire was held in place by a plastic stop. However, this stop was molded directly over the wire, and more importantly over a knot in the wire. This stop however is molded to fit the headphones and so I wanted to keep it intact when I actually removed the wire. I ended up actually drilling the wire out and then prying the remaining bits (including the knot) out with a screwdriver, which left me with a clean stop to thread the cord though.
After the wire was heatshrunk in place, it was a relatively simple matter of soldering it to the speaker and reassembling everything.
In the end the repaired wire was shorter than the original, but this isn't such a big issue