The leaking substance is potassium hydroxide, a strong alkaline (alkaline battery....get it?). Use acid/vinegar to neutralize that, and clean corrosion.
Why Do Batteries Leak?
If the mess isn't too bad, and the contacts are not very corroded you could possibly go after it with Q-tips or a small toothbrush (some kind of appropriately sized brush), just....barely....wetted with vinegar. Work carefully, possibly upside down so liquid can't pool in the bottom. Finish with a weak bicarb solution cleaning (neutralize the acid), then a fresh water cleaning.
How Do I Clean Battery Acid Leakage in a Remote?
If you don't want to take it on, you have the offer above.
As you know now, don't leave alkaleaks in a device for a long time (I've grown to hate them, does it show?). FWIW, a rechargeable NiMh eneloop will do as well (30 vs 35 hours) and has no chance of leaking, plus you can recharge it. Excellent battery!
What type of batteries can I use in the Petrel? - Shearwater Research
This is great. Thanks, Fritz!!