How do I get my keyboard to turn on after removing corroded batteries?

The top battery exploded but I used baking soda and water on a q-tip to get the cap and two batteries out of the keyboard. I did some cleaning, dried the keyboard, but now it won't turn on after inserting new batteries. Any solutions?

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The corrosion and/or baking soda/water solution may have gotten inside the keyboard, to the circuitry. While it's possible to separate the Apple Wireless Keyboard for assessment, it's not an easy job.

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