Why isn't my iPad charging?

It charged fine before, but yesterday I let the battery go all the way down to empty, then plugged in the charger. However, the iPad never starts back up, not even with the Apple logo. The screen is black, with the white battery icon and a thin red line inside it. I even left the iPad connected to the charger overnight, but it still shows the battery icon, and never charges.

I suspect it has to do with the cables, as they're pretty old (they were originally for an iPod touch, 3rd generation), but so is the iPad (it's around 2-3 years old, and it was a hand-down from a friend).

How can I get my iPad up and charging again? Do I simply need to replace the old cables with a new one, or is it more to do with the iPad's internal battery? I dropped it on a hard surface, and while there was no obvious exterior damage (other than a small crack on the screen) and the iPad working well, there was probably internal damage to the battery, which may be why it's not charging.