To marry her you have to find out why she's sick.
Go to the library and check out the bookcases. You'll find a book about the legendary Cursed Stone. Head back to the Library after rescuing "Ceci" from Clements Cave. They'll thank you for saving her and Cecilia will mention that the book about the cursed stone is scarey. In the evening when Felicity is upstairs at her father's house, talk to her about the stone. Felicity will hand the stone over to you. Make sure you have an available slot in your Rucksack to receive the stone. You will need to have at least 3 FP with Godwin and have been given permission to enter his house after 6:00 pm. Take the stone to Doctor Edward. He'll examine the little rock and confirm that it is indeed a Cursed Stone. Edward will dispose of the little rock for you. Return to Felicity and talk to her. Tell her what she received was an evil Cursed Stone. Now that she no longer has the item in her possession, her health begins to return. Once you've completed this meet up with Felicity on the holiday at the Plaza between Noon and 4:00 pm. You can then propose to her if you've met the basic marriage requirements.