Mr. Nichols, whose long career experienced a few dips but no serious lulls, was working right up until the end. He directed hit revivals of Death of a Salesman and Betrayal in 2012 and 2013, respectively. He won a Tony Award for his direction of the former—his eighth Tony for directing. (He won two additional Tonys for producing.) His more recent film was "Charlie Wilson's War" in 2007.