Does it charge when it is turned off? If yes, then the problem may be the charger cord. When they start to go they don't put out enough power to both run the device and charge the battery, but will charge fine when turned off.
However, your mentioning that it won't hold a charge wouldn't be the charger cord but the battery itself. Over time they do go bad and will start to hold less and less of a charge. Making it seem like they fully charge faster, but that is only because less and less of it is able to charge. Which then makes it die out faster.
If you have another charger, or can borrow one that is like yours, I would test that first. If it resolves the problem, you should get a new charger. If it doesn't help then I would try a new battery.
In the end, you may even need both.