Update, almost one year to the day after last posting.
I was not able to get my Klipsch X10i with the frayed jack plug repaired, due to the cable being so thin and fragile. I therefore sold the headphones as spares or repair on eBay and, because I liked the Klipsch sound so much, purchased another pair of Klipsch X10s (not X10is) from a forum member here.
Unfortunately, it seems that buying another pair of Klipsch was a bad idea; the build quality is truly awful, because my Klipsch X10s have now broken, this time due (i) the black part of the earbud coming away from the gold bit (which I managed to resolve with a glue repair); and (ii) the cord fraying near one of the earbuds (which I suspect cannot be repaired).
Here are some pictures of the damage:
I've lost sound in the damaged earbud.
Most annoyingly, I have no recourse or prospect that Klipsch will repair these headphones free of charge.
I will not be making the mistake of buying Klipsch for a third time.