Unforunately, it seems this isn't a very easy to fix problem. I looked around a bit and it seems that this is just a common quality problem. Appears to be a pretty common story on this generation of Logitech Speakers, unlike the Z2300 and z5500 days, as many have said the quality of the cables and parts have just gotten worse.
I'd say instead of trying to fix it yourself, just claim warranty. Logitech warranties the z623 for 2 years, so you're warranty should still be good in contrary to what you say. I would just send it in for warranty. It costs money yes, but its better than having to buy new speakers.
OR if you want to buy new speakers I would get some Klipsch Promedia 2.1 or Swan M10s if you can find them.
Here is a linky
to the warranty terms for your speakers.