I have been involved with Recruitment at UW for a couple of years and thought I would add in my two cents/reiterate what others have said.
VandalSquirrel is right - housing is definitely at a premium at UW and your daughter would be taking a chance by not securing a dorm room and going through Recruitment. However, I know first hand that the women on Panhellenic do everything they possibly can to ensure that a PNM going through matches to a chapter (ie has a sorority to live in).
As to legacies, there are quite a few that come through Recruitment, however I would venture to say that maybe one or two chapters on campus have enough legacies to fill up to quota. That's not to say that chapters automatically take their legacies - it's more along the lines of looking to see who is a fit for that chapter, legacy or not (realize this is a general statement on what I see from the UW greek system as a whole).
Now onto the not-matching portion - as I stated above, the women on Panhellenic really do try their best to make sure PNMs are matched with a sorority, though a lot of it depends on if the PNM is willing to accept only certain sororities or any sorority. The PNMs who don't match to a sorority are almost always the ones who "suicide" or "single intentional pref" - meaning they put down fewer sororities on their pref card than the number of sorority pref parties they attended. In fact, I can't think of a single case in the past couple of years where a PNM who maximized her options was not given a bid.
Lastly, if you daughter decides greek life is not for her halfway through Recruitment, she will not be stuck with nothing to do the week before school. The University sponsers Dawg Daze, a week-long series of events for incoming freshman. You can find more information here:
http://http://depts.washington.edu/dawgdaze/index.php
In my opinion the UW greek system is a wonderful community with some amazing women - good luck to your daughter (and you!) if/when she goes through Recruitment!