Try to track down exactly where the break mightbe, if it is in the connection, or in the cable somewhere, that is the best advice I can give you. To do this, get something/someone to hold to spots on the cable still and shake inbetween them, moving your way up and down the cable a little at the time and see if it affects things, untill you find something. If you dont find anything check the connections by turning them. You can lightly shake them but note that shaking them may make things worse if the problem is in the jack itself, and probably wont help things at all. Also be sure to check where the cable enters the speaker encolsure. I have learned not to buy Logitech due to their habits of using proprietary connections on their speakers myself
If the problem is in the connection, one thing you can do is get some electrical contact cleaner(Check your local hardware store or electrical supply store) and some steel wool and clean the connector and try again. If this doesnt help then the problem is probably t jack, which to replace requires a good knowledge of soldering most of the time and isnt really reccomended.