Witwyn has grown concerned for Éowyn's state of mind, and invites you to have a look around the lady's room while you deliver her next meal.
'I am sure you think Lady Éowyn is bearing all of this very well, but I believe she isn't happy. A servant notices things, you know.
'Do you want to see for yourself? Deliver her meal to her room and have a look around. Tell me what you think!'
Éowyn's serving of food can be found on a table near the throne of Meduseld. Her room is to the left of the throne.
You should bring Éowyn's food to her room to await her.
Éowyn's room is in Meduseld.
You should examine Éowyn's room for signs that Witwyn is correct about the Lady's gloomy mood.