I appreciate that this is a common TL, but it can't hurt to give it a fresh go. In this scenario, Nixon avoids being hospitalised after injuring himself on a car door and wasting two weeks. In addition, he looks better and healthier in the first debate with Kennedy. As a result of the extra fortnight of campaigning and the better performance, Nixon edges Kennedy.
From the New York Times, 9th November 1960:
Nixon Victorious
VP Edges Victory Over Senator John F. Kennedy
Vice-President Richard M. Nixon defeated Senator John F. Kennedy last night in a battle of a closeness not seen for decades.
The high drama of the night came when Mr. Nixon took Illinois by just 2,000 votes after a recount. New Jersey was also secured by a razor-thin margin. Hawaii and Missouri also just fell into the Republican camp.
Senator Kennedy has conceded defeat and has returned to Boston to commiserate with his family.