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I usually just go to the nicest airline lounge where I have status; If you don't have status, you can pay for one day. It's not cheap but probably less than renting a room. I've passed out on the sofa for several hours and noone has ever bothered me.
I know this won't really help because its not near LAX, but I heard of this place near Koreatown where people go when they are too drunk to drive home. You can rent a sofa for a few hours and sleep. It's supposedly connected to a nice day spa too. Random, I know
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