I have never stayed overnight in that area but I did find a link to a discussion in a forum. Basically nap at the station or get a nearby room. Here's the link: https://discuss.amtraktrains.com/index.php?/topic/16102-any-advice-for-where-to-sleep-near-king-st-station-in-seattle-please/ Good luck!
You can pass out in the waiting room if they stay open all night. I doubt anyone is going to object if you are ticketed to ride. If however they wont let you crash in the waiting room perhaps there is a doorway or the benches just outside the front door.
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