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When batteries leak, it is so corrosive that it often permanently damages electronics. The best thing you can do is use a scrub brush and some rubbing alcohol. A toothbrush and sandpaper will work just fine. Clean up the battery compartment really well. A lot of times the corrosive white powder you see will prevent contact with batteries, and it won't come on.
If you get it cleaned up really well, and are certain that the batteries are making contact, I think the battery acid has done permanent damage.
Do you see any corrosion on the circuits?