Yeah... Unfortunately, that scene is programmed to only proc after you saved the hive but before you recruit Azzy. So if you recruit her at any point, you're effectively locked out of that event for the rest of the playthrough. However, there IS a way to manually start the event... If you're not against a little console wizardry, that is.
Whilst playing the browser version of the game, hit F12 to bring up developer tools, and punch the following command into the console:
AzyrranXmas();
This will force the game to run Azzy's holiday event, regardless of what you did or were doing beforehand.