According to the Captain's Log (the book in the wheelhouse), Dagny has a key. Ask her for it --
-- and she just hands it to you without first making you find her a pineapple daiquiri or solve a complex trigonometry problem. How weird is that?
Use the key in the radio to talk to a friend of Magnus and learn some more about his disappearance.
Once you do something for her, of course. Ah, there we are -- back to Nancy normal.
How do I get to the lighthouse?
How do I fix the lighthouse light?