Everyone has something to offer Dream in return for the key to Hell, but the demons bring the most leverage: they would trade Choronzon (a demon Morpheus once had to duel) to torture as he sees fit, and more importantly they offer him Nada (and threaten to devour her soul if their bargain is refused). Ultimately, though, Dream grants the key to the angels, who expressly do not want it but have been commanded by their Creator to oversee Hell in Lucifer's absence. It makes sense: no one who wants Hell for themselves is fit to tend it.
This also discloses some very interesting truths about the duality of existence in Gaiman's fiction: for Heaven (err, "The Silver City") to exist, its antithesis must also exist. Since Heaven and Hell are reflections of each other, both are needed and thus Hell must be opened for business once more. Ponder the fate of the angels assigned to Hell's governance; it's fascinating that Heaven must manufacture its own rebels, embittered by the task given them, for its own maintenance.
Of course, Morpheus eventually does battle Azazel for Nada's return--and win. His reunion with Nada--after what has to be the world's weakest apology for condemning her to ten-thousand years of Hell--is brief and bittersweet: he rejects her offer of becoming mortal to be with her, she rejects his offer to make her a goddess, and she decides upon being reborn as an infant in Hong Kong. Loki manages to trickster his way out of returning to his Asgardian punishment, and Lucifer learns to love life on the beach. Oh, and along the way Gaiman drops a piece of canonical lore that will become the fulcrum of the saga: the current Endless are but aspects of what they represent. They can die, and can abdicate responsibility for their realms as Lucifer did, but there will always be another aspect to replace them. The choices made on the path of destiny have consequences, even for the Endless.