I've had that problem with two kindles so far. They replaced the first as it was on warranty but refused to replace the second when it broke as well because it wasn't within the warranty of the first kindle - that broke. My suggestion, get a refund if you can and buy something better or, if you can't, get it replaced ASAP before they decide you're past your warranty date. Such a shame... I loved the kindle but the charger port is c%&p.
I had a similar experience. Amazon replaced the 1st one. Two weeks after I got the replacement, the same thing happened. They are replacing it with a Kindle Fire 2nd generation, but I have a feeling it will have the same issue eventually.