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
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