I inadvertently left some Duracell alkaline batteries in the Balance Board. Sure enough, they were already starting to leak – damaging the battery contacts in the Balance Board.
It turns out there is a relatively easy way to remove the gunk from the leaky Duracells: they are alkaline batteries, so a mild acid (household vinegar) should do the trick. After disassembling the Balance Board, carefully remove the corroded metal contacts from the battery holder and drop them in a small jar with household vinegar:
Watch the corrosion dissolve; if needed, use a toothbrush or Q-tip to brush the last bits of gunk from the contacts. Rinse with water, and allow the contacts to properly dry before re-assembly.
P.S. It seems that Duracell batteries are quite prone to leaking – quality sure went downhill over the years. I’m replacing all of them to prevent further damage.