niobe - Nancy Drew: Labyrinth of Lies

when does Niobe ask Nancy for permission for the key from the locked display case

The game is nonlinear, so I can't tell you when that happens exactly. I suggest looking at the GameBoomers walkthrough if you haven't already. And if you are playing in Junior/Amateur mode, you should look at your task list to see what still needs to be completed, which might aid in helping you ascertain what things you should try next.