My old computer had no working battery and the AC charger didn't work on it so I couldn't transfer the data.
I hope you still have the old computer, because really that sounds like a minor issue, you could easily get a new battery and AC charger, but even cheaper you can get a small adapter to connect the hard drive from your old computer to a USB port on your new one and copy all your old files over yourself.
Just to give you an idea of what you are looking for if you go this route:
Vantec USB 2.0 to SATA (Serial ATA)/IDE Adapter