OP,
It sounds like the plug or jack it broken. I had the same problem on a NOOK, and it turned out that buying a new cable solved the problem. If it is the jack (inside the Kindle), this may be expensive to repair.
There are a couple of computer repair shops that might be able to help. I used MMD (441 just south of 466) recently and was very happy.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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