Soak the area in vinegar or lemon juice for a bit. Then rinse with warm running water. Follow up with a toothbrush and baking soda, another warm water rinse, then (finally) a rinse with distilled water. Blot up the excess moisture with a paper towel and let it dry completely. Remove and replace any nearby parts that show signs of damage (corroded IC legs or pads), and make sure to thoroughly clean underneath them.
A new battery isn't a must.
