It sounds like you have two problems. One with the headphones and one with one of your cables.
First - sound in one ear only is probably that the connection between the jack and headphone speaker is open. Either lack of tension in the spring connector in the jack itself where it touches the plug - (gently try to move plug side to side / up and down whilst inserted in jack and listen for sound in speaker side not working), wire off jack terminal, wire broken between jack and speaker, wire off speaker terminal or faulty speaker. The convention is/was that the "tip" of the plug connected to the left speaker and the "ring" (next connection on plug) connected to the right speaker and the "sleeve" was the common (or earth or return) for both speakers. But sometimes manufacturers do whatever they want.
You will have to open the headphones up and using an ohmmeter do a point to point test of the wiring path to locate where the problem is.
Next - the cable that gave no sound could have a faulty connection say for the tip (left speaker) and if the headphones had a faulty connection for the right speaker then you would end up with no sound in either ear, which I think you said happened. Of course it could be the other way around or it could be a faulty "sleeve" connection in the cable as well. You could try inserting this cable and the gently flexing it just past the plug at either end of the cable and listen for any sounds. This may tell you where the problem is if the wire has just come loose in the plug. Then again you may not hear anything.
Hopefully this has been of some help