Or perhaps Elza retrieved the Braclets for her, and left the CK because she didn't know what it was.

Or perhaps it was hidden in a different part of the compartment.

Finally, remember what Rand went through in the cleansing while trying to weave Saidar? He nearly died, as did Nynaeve. Semi possibly didn't think she could have Rand use the CK and live through it. It might even have been enough Saidin to destroy the DB itself from the sheer amount of it.