There is another thing to consider in this situation, and that is
Eowyn's knowledge of the interest of
Grima Wormtongue. She was in close proximity to the King who was becoming more and more frail. She was, if you will, his primary care-giver.
Grima was also almost constantly in the King's presence; he could not afford it to be otherwise lest someone wrest
Theoden's will from his keeping. That means that day after day, Eowyn was subjected to Grima's "attentions" whether they were actually stated or not. Women know what it is like to be forced into close proximity with someone who "undresses" them with their eyes and whose every utterance may have a double meaning. Furthermore, as Grima grew in importance to Theoden, the King might not have been averse to a "match" between his trusted counsellor and his niece, especially as he had grown to mistrust
Eomer. This sort of sexually charged situation can make the man/woman relationship into something ugly and degenerate in the mind of a maiden like Eowny.
Then, into this intolerable situation comes
Aragorn. He is a breath of clean, fresh air: courageous, manly and courteous. No wonder the wounded, beset Eowyn casts her eyes on him as a means of fleeing her trap. After all, if he is in fact, the true
King of Gondor - and she can interest him in her person - Theoden would gladly betroth her to him and move her forever beyond the reach of Grima and Theoden!
In the Houses of Healing, Grima is gone and, for all intents and purposes, so is Aragorn. The King is dead and her brother is King, so she need not fear that she will be unworthily wed. She has proven her strength and courage to even
her satisfaction - after all, her name will live forever in the annals of
Middle Earth as the maiden who slew the great Captain of the
Nazgul. But after all that has happened, she still is "unwell"; her inner despair has not healed. The release of death has not come and the clash of arms and victory have proven less satisfactory than she had believed would be the case.
Into this "emptiness" comes
Faramir. She knows his story and is drawn to his courage as she was drawn to Aragorn's. Also, he is fair of face but even more importantly, he is not "unmindful" of her as was her uncle and brother (except for their familial love, Eowyn of
Rohan was still a woman and therefore of less importance than a male child would have been), nor is he "uninterested" in her as a woman as was Aragorn. Faramir is drawn to her as was Grima, but his "affection" does not fill her with rage and self-loathing. Furthermore, he treats her very much as an equal, a warrior proven in battle and worth of respect. This is new - and quite heady - for remember, Eowyn is yet very young. All of these conflicting emotions and impressions especially superimposed upon her physical suffering, would have the effect of acting
very quickly and, hence, the "romance" between the future
Prince of Ithilien and
the Golden Lady of Rohan is born.
The situation may appear to be "convenient", but actually
Tolkien does develop the matter quite fully if one is paying attention to the story as it progresses.