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Based on the description, this appears to have happened during Houdini's 8 week run at Princess Theater in Chicago in early 1926. A bad year for accidents!
Houdini wasn't the only magician to have a mishap with the Needles. In 1946 an amateur magician named Stanley McAuley garnered headlines when he swallowed a needle during his own performance of the trick. It took two operations to retrieve it.