A couple of thoughts: Alkaline batteries leak alkaline, so baking soda (already alkaline) is not a neutralizer. Use mild white vinegar, an acid, to neutralize the alkaline. The neutralized alkaline will be a neutral salt, and still needs to be removed. Try duct taping a q-tip to a pencil. Clean carefully and thoroughly.
When alkaline batteries discharge, they swell, so don't wait until the batteries are dead to change them.
If and when the old leaky batteries can be removed, replace them with rechargeable nickel metal hydride (NIMH) batteries. They are very robust and chemically more stable than alkaline batteries. Therefore, far less likely to leak.
Good luck to all!