RORY: Taylor’s wigging.
LORELAI: I know. He’s been sitting there like the final days of Dick Nixon for almost an hour.
Richard Nixon (1913-1994), 37th President of the US from 1969 to 1974. The Watergate Affair, the name used to describe the secret and illegal activities undertaken by members of the Nixon administration, was brought to light by reporters Carl Bernstein and Bob Woodward in The Washington Post in 1972.
Nixon had hoped to weather the storm by refusing to leave, but impeachment hearings against him opened in May 1974. With loss of political support, and the near-certainty he would be impeached and removed from office, Nixon resigned on August 9 1974.
Lorelai may be specifically referring to the 1976 non-fiction book, The Final Days, by Carl Bernstein and Bob Woodward, describing the last months of Richard Nixon’s presidency. It was a major commercial success, and was made into a television movie of the same name in 1989, with Lane Smith as Richard Nixon.