“Sheila P. Burke offered her resignation as deputy secretary and chief operating officer, after working for the Smithsonian for seven years while earning more than $1.2 million in six years for outside duties, a Washington Post review showed. … Burke’s resignation came on the eve of an independent report that sources said would criticize her extensive outside activities, including highly compensated corporate board seats, academic appointments, a federal commission that oversees Medicare, and numerous nonprofit organizations.”