The hospital I work for has at least a hundred RN positions available. If you want to work out here, you'll be able to find a job.
You might think about checking with some of the other hospitals out here and schedule some interviews with them while you are here. Bus transportation from the U District to First Hill is pretty easy with a transfer downtown. Look at
Swedish Medical Center, Seattle, Washington | Swedish Medical Center / Seattle,
http://www.virginiamason.org for the two hospitals on First Hill, near downtown. (Harborview is also here, but you're probably going to the same HR department that hires for them, Children's and UWMC.)