This is one of those episodes where Columbo does a deep psy-op test on his potential perp right out of the gate. With Nora, his test was the following, and I believe was the moment he was really on to her.

If Nora is innocent, and had just recently learned of the horrific way in which her beloved secretary had died, she is going to behave and react in a certain way to Columbo's questioning. When Columbo rains-down praise on her when he enters her home at the studio, she responds in a way that tips her cards by presenting a demeanor to Columbo that she isn't really all that traumatized. "Never thought you'd meet a legend, right?..", etc.

Then, of course, Columbo doubles-down (verifies) this test by asking her to call his wife for a chat, out of the blue. Weird right?

Well, we all know if Ms. Chandler was actually innocent, how she would have reacted to a random detective's request for her to actually pick up the phone right there and CALL some unknown person for a chat about nothing, in the midst of such awful circumstances. She's snap at Columbo for asking such a crazy thing right there, and tell him to get to work trying to find out what happened to her secretary. But, she actually plays along with Columbo's bizarre request, leading him to believe she may be on the other side of innocent.