I'm in the same type of predicament you're in, but perhaps I can be of some assistance. I'll be graduating this Spring (well, a few months away now), and in a year will be applying to graduate schools in hopes of obtaining a graduate degree in pre-modern Asian history. I'm rather choosey in where I look, though, being that I'm unwilling to move too far away (east) of where I grew up. Many places I've looked have very well developed Asian Studies programs, so here is a list of a handful of them you may want to look into. Be aware that these are classified with the objective of locating pre-modern Asian graduate history programs. However, in general, because of the scarcity of these programs, this usually means a good Asian studies program as well.
University of Hawaii
- My first choice right now, a GREAT Asian/American exchange program. Its location makes it prime for just hopping over to Asia as well.
University of Michigan
- A great Asian history program, very selective but very very accredited.
University of Minnesota
- Another great Asian history program specializing primarily in Tang history and art. Also the home of one of the more prominent Asian history journals published in the United States.
University of Washington
- As you mentioned, a good option for Asian studies. I live a few hours from here and I can say that it is in a great location with a good Asian population. Having the option to engage in conversation in whatever language you're wanting to speak is a necessary for becoming proficient in Asian studies.
Michigan State University
- Another good Asian history program, though not as accredited as U of Michigan. The focus here (in the history department) is primarily on Song through Qing dynasty China, but it also has a very well developed medieval Japanese history program.
University of New Mexico
- A pretty good Asian history program, but the language department leaves something to be desired.
University of California - Berkeley
- A very good college with a very good and well accredited Asian Studies program. Not my cup of tea due to location, but very prestigious and worth looking at if you're willing to live there. (Yes, I know you mentioned this one)
University of Chicago
- Again, you mentioned this one but it's worth posting a link to. Very selective, very prestigious, very good. About as far away from Asia as you can get, but somehow manages to have an extraordinary Asian Studies/history program.
This is just a small selection, and only includes those places that offer Ph. D.'s, but hopefully it will help some. As far as Ivy league colleges go, they're a bit out of my league so I haven't done much research on them. Maybe this should be stickied? I've been looking for a thread like this for months.