I'd say Marc Eliot. I know it isn't a popular choice here but it seems to be most comprehensive and best researched of the ones I've read. That doesn't means it's perfect, balanced and without errors but at least in most cases he shows where he got his information.
I started to put together notes on some of the other books, so I'll post what I have so far.
"Flying High" by Laura Jackson
It's an easy read and she writes like a fan. In places it read like a novel with the author describing what the characters were thinking and feeling when she has no way of knowing this. The source material appears to be old interviews and other books rather than any new interviews or research and there are some glaring errors (how could she think that the Desperado cover shoot took place in London?).
"Eagles Taking it to the Limit" by Ben Fong-Torres
Possibly the newest of the books, being published in 2011. The author worked for Rolling Stone magazine when the band was in its prime, took part in the famous soft ball game and interviewed the band during the Farewell 1 tour. He makes use of old stories, record reviews and, presumably, his notes and memories . He was there – sort of. There are errors (at one point he mixes up Randy and Bernie) and there isn’t much new content but it’s an easy read and maybe as good an overview of the band’s history as any. And then there are the photos…
Over a hundred of them, mostly colour, many full page. Even Bernie gets two full page photos just of him. If you’re tempted to buy a tour book just for the photos, then this offers much better value. I paid just £5.99. Seriously, the photos are good.
"Hotel California" by Barney Hoskins
This is the one that opens with five naked young men in a sauna.
I really enjoyed this. It has a broader sweep covering The Byrds, Buffalo Springfield, CSN&Y, Joni Mitchell, David Geffen which all gives context and perhaps a better perspective. Hoskyns interviewed lots of people, including JD, Jackson and Bernie. Lots of good anecdotes.