There is no aux cable on that model. There's the remote cable that goes to the control head and a jack for the satellite speakers. And of course the power cord. And the cable that goes to the source.
If you wiggle the cable going to the control head and it temporarily fixes the problem more than likely there's a wire broken inside the jacket. It's a real bear fixing that. You have the volume pot, headphone jack, and power switch. That's at the least 8 conductors of 24 or 26 ga wire. That's 16 solder joints you have to redo.
If it's the speaker wire you're referring to then possibly the jack on the back panel is worn out. Takes longer for the iron to heat up than to do the actual replacement.
Not being an ass about it but if you gave the current price of $49.99usd for it and it's worked for 4 years, I think you got more than you money's worth out of it. I'd hop on Newegg and get a replacement.
Logitech Z313 Speakers - Newegg.com