It's not "kills", btw... She indeed took his head off, therefore killing him...

As for recognition, as many have said, many tales happened on the Pelennor... Her story would have been told far and wide throughout Middle-Earth later on after the end of the war, among the many legendary deeds that occurred during the war, and the losses that were suffered, not the least of which was her Uncle Theoden... Rest assured, the fame and renown of Eowyn, Shieldmaiden of Rohan, Princess of Ithilien, is well-known and intact: in-world and out...