864
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 original was "why didn't the eagles fly to mordor and drop the ring in the top of the volcano" which was funnier the first time than the thousandth time. You're right that these witty workarounds are only funny the first time and tire quickly.
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 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.