Kash Patel flew to Miami on Air Force One last weekend to watch an Ultimate Fighting Championship event, wearing his signature wraparound sunglasses — at least the second time he has gone to a mixed-martial arts fight as F.B.I. director. Days earlier, he showed up at two N.H.L. games, grinning in photographs with the hockey legend Wayne Gretzky. At one, in Washington, Mr. Patel, who has played the sport since he was a child, was spotted in the owner’s suite as he watched the Capitals player