Because kendamas are made from wood, a natural material, it is normal that over time chips or splinters may occur. Certain tricks like Insta Flips and Taps can increase this possibility, as well as playing over hard surfaces like concrete.
It is very easy to repair a chip using wood glue or superglue to put the pieces back together. Use a small bit of glue to re-attach the chip and use a rubber band or clamp to keep the piece in place overnight while the glue dries. You may need to sand down any glue that seeped out after it dries. This should make the kendama fully playable again, and the glue might make that area stronger than before.