Are you looking for cheap, or nice? Will you have a car? Also, will your interview be at the UW hospital, or somewhere else?
I stayed at the University Plaza the first time I went there. It was a dive, but fairly close to UW Hospital. The second time, I stayed at a Days Inn on 7th avenue. Another dive, but cheap and downtown and near a bus route to anywhere. Walking distance to Harborview. My last day there I stayed at the Mayflower Hotel, which is also downtown. Verrrry nice, and not hideously expensive (although it was a concierge-connection-deal, so I don't know if you'd be able to stay as inexpensively).