Gwaewar, King of All Birds was a son of Thorondor and the father of both Gwaihir and Landroval. It was Gwaewar who Gandalf healed of poisoning in his youth; in return for the Istar’s service, he and his kinsmen saved Thórin & Company from the Orcs and Wargs of Goblin-Gate.
Gwaewar originally was an early name of Gwaihir in J.R.R. Tolkien’s first drafts of the Lord of the Rings. Later however he discarded it. The Name is therefore reused here for Gwaihir’s predecessor in the Hobbit. As Gwaihir was said to have carried Gandalf only two times, the Eagle King in the Hobbit must have been his predecessor, possibly his father.
However, Gandalf also states that Gwaihir and Landroval were the mightiest of Thorondor's descendants. So the way this can all be connected with Gandalf's published statements being correct is that Gwaewar was the King of all Birds at the time of the Hobbit, was father to both Landroval & Gwaihir, carried Gandalf during the time of the Hobbit, but finally--died somehow between the time of the Hobbit and the time of the Lord of the Rings.
All the above listed points would allow for all statements that Gandalf made in the published text to remain true.