First, look at the battery contacts of the broken controller (the metal parts where the battery connects ). Are they clean and shiny, or do they look oxidized, dull, blue or green? If they are not shiny you can try cleaning them with a scotch brite pad, or sand paper if it's really bad. If there is old battery leakage you can try wiping it a bit with hydrogen peroxide first before scrubbing it till it's shiny. Also check to see if the battery contacts are bent in so that they do not have good contact with the battery; if that is the case you can try bending them outwards for better contact. If those don't work, you may need to open the controller and have a look around with an ohm meter to check for continuity. It might be that it was dropped in a certain way or left in the heat, and a solder connection may have come undone and may need to be re-flowed.
I've also heard of the buttons in this controller getting stuck, and supposedly hitting it against something(not too hard) could un-stick the buttons. If you open it you can try to take a closer look at the button system.