From LotR Book VI,
The Field of Cormallen, it seems to me that Gwaihir's brother Landroval, rather than Gwaihir himself, was the King of Eagles mentioned in The Hobbit:
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There came Gwaihir the Windlord, and Landroval his brother, greatest of all the Eagles of the North, mightiest of the descendants of old Thorondor
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If the rules for Eagle royalty were similar to those observed among earthbound bipeds, it would make sense for the King not to go venturing forth far and wide from his own realm that often (unless he was leading his host to war, as in the Battle of the Five Armies and the Battle of the Morannon), whereas Gwaihir, as the King's (presumably) younger brother, would have had more freedom to deal actively with matters of foreign policy (such as rescuing allied Wizards from towers or mountain tops). He may even have been acting as the King's emissary, paying Gandalf for a service not given to himself but his brother, when Landroval's other royal obligations kept him from coming to Gandalf's aid in person.