You need to complete all events and actions with her. Clicking on the hint itself will tell you more. But in short, keep her happy and full, talk to her, give her special items, and do actions.
At what point should I give her the sword? I'm stuck at this point and have no idea what to do :(
The sword is needed to do the “Kiss” action with Asuna. Try different ways and you'll get it done.
