854
But first they would need to invent some tape
(i.imgur.com)
They could have fitted the whole ring / tape / mouse assembly into a small paper bag Aragorn could have kept it in his jacket and fed it little bits of lembas on the way how lovely x
The answer is that the eagles are theologically angels in LotR and both sapient and very susceptible to the Ring, and having a flying Sauron-replacement is not an improvement.
I thought it was that the eagles were basically demigods and they didn't originally help frodo for the same reason we don't regularly help carry ants back to their mound with food but then after Gandalf came back he was able to convince them
I just assumed that sauron had some means of stopping eagles flying to mordor
He's got the Nazguls riding the Fellbeasts as well as the fact that if the big ass eagles flew directly at Mount Doom, everyone would see them and orcs would be swarming the mountain before they could land.