You can probably do one of a couple things here:
# Start by using a fine grit, wet/dry sand paper (about 600+) and sand off the corrosion.
# Disassemble the controller and get to the battery terminal. Remove it (desoldering or other means) and soak it in white distilled vinegar. Brush it down to remove any of that blue corrosion and then soak it in rubbing alcohol and clean it off. Solder it back on and see if it works.
# Purchase a busted controler off eBay and replace that particlar piece with the one you got off eBay. You may need to desolder it and put it onto the new one.
# Finally, if the corrosion has affected the board. Getting a busted controller off eBay and replacing the entire board may be your final option.
Without pictures of the actual damage, these are the best options I can suggest to get the device running again.
Let us know if any of these suggestions work and please rate my answer if it helped solve your problem.
Good luck!