Could be helpful to others in the future if you would list the name of the hotel....
Good thinking. It's the American Hotel Hostel, 520 South King Street, International District, Seattle, WA 98104. I opted for a single room (not shared) but with a shared bathroom. There is a coin-op laundry on the premises so I'll be doing laundry there too.
Two nights cost a total of about $170, complete. I'm happy with that. I only need it for sleeping, showering, and doing laundry.
It's walking distance to the station.
Once I've stayed there, I'll try to remember to report back here on how it is.