When you say you get no power from the power cord, are you meaning that there is no signs of life whatsoever? that is no LED's are illuminating, if this applies then are you sure the power cord isn't actually defective? as if it is then thats why the battery is failing, as its not being charged enough to have sufficient power to activate the camcorder.
Of course if nothing mentioned applies, then its a case of a more detailed investigation being made with the aid of a voltmeter, needless to say the casing would have to be separated to do this.