If Earth Kingdom Jailer leaves the battlefield before its triggered ability resolves, the target artifact, creature, or enchantment won't be exiled.

Auras attached to the exiled permanent will be put into their owners' graveyards. Any Equipment will become unattached and remain on the battlefield. Any counters on the exiled permanent will cease to exist. When the card returns to the battlefield, it will be a new object with no connection to the card that was exiled.

If an Aura is exiled this way, its owner chooses what it will enchant as it returns to the battlefield. An Aura put onto the battlefield this way doesn't target anything (so it could be attached to a permanent with shroud, for example), but the Aura's enchant ability restricts what it can be attached to. If the Aura can't legally be attached to anything, it remains in exile for the rest of the game.