As he warmed up the crowd at Studio 8G on the evening of Jan. 10, Seth Meyers had a special request. Before the cameras rolled, he told those in his audience at “Late Night” that, though they were used to seeing him interview celebrities and newsmakers with whom he often agrees, he wanted them to be respectful of a guest “who does not agree with me on a lot of things.”

Soon, he would talk with Kellyanne Conway, who would become the White House counselor to President Donald J. Trump after serving as his campaign manager. Mr. Meyers, the literate, quip-firing comedian and host of NBC’s “Late Night,” had spent the preceding days mapping out his conversation with Ms. Conway, a steadfast and unflappable spokeswoman for Mr. Trump.

Then, about an hour before “Late Night” was taped, the world learned that Mr. Trump had received an intelligence briefing on a possible Russian attempt to compromise him, and Mr. Meyers had to decide how to address this breaking news with a key member of Mr. Trump’s team.

This was all occurring days before Mr. Trump was sworn in as president; before his White House press secretary, Sean Spicer, gave a combative briefing in which he falsely stated that Mr. Trump drew “the largest audience to ever witness an inauguration”; and before Ms. Conway said on “Meet the Press” that Mr. Spicer was merely offering “alternative facts.” (Even as this sentence was being typed, Mr. Spicer was telling White House reporters, “Sometimes we can disagree with the facts.”)