To finish Asuna's story, you must go through all the talks and actions with her. If she is in a good mood and not hungry when you talk to her, her conversation event should start, if available. If she is in a good mood, but the event does not start, then you need to do the next action with her. Usually, there will be a message if you need to buy a special item for her, but it is better to buy them all as they appear in the store.