I'm trying to find the average annual household income in Korea, for
Seoul alone and for the rest of the country other than Seoul.  In
other words, I'd like to know how much the average household living in
Seoul makes in a year, and how much the average Korean household that
does not live in Seoul makes in a year.

This is not precisely what you're asking for, but you might find it helpful.

The Bank of Korea released a report in December 2004 which provided
this information:

"The nationwide average of per capita income (per capita GRDP) was
recorded at 12.25 million won, and Ulsan was the richest city in Korea
with a figure of 26.87 million won, followed by South Chungcheong
Province (14.74 million won), North Gyeongsang Province (13.96 million
won), North Chungcheong Province (13.42 million won), South Jeolla
Province (13.34 million won), South Gyeongsang Province (13.02 million
won), Gyeonggi Province (12.82 million won) and Seoul (12.75 million
won), all of which are over the average.

The per capita income of Seoul, Gyeonggi Province and South Gyeongsang
Province ranked in middle despite their large economies, as their
populations were relatively larger compared with their output.

Daegu was the poorest city with a per capita income of 8.01 million
won, while Gwangju recorded 9.33 million won, following by Incheon
(10.97 million won), Jeju (10.13 million won), Daejeon (9.8 million
won), North Jeonla Province (9.70 million won), Gangwon province (9.62
million won), Busan (9.58 million) and Gwangju (9.33 million won), all
of which recorded below average.

The per capita income of the last three regions that accounted for
70.0 percent of that of the top three regions in 1985 dropped to 61.8
percent in 2002, indicating that regional disparity has continued to
expand gradually."

Seoul Metropolitan Government: Economic power of Seoul 
http://english.seoul.go.kr/today/news/newsclip/1216075_3675.html

Try this site:

http://engshopmall.nso.go.kr/

It's a site that sells statistics about the Korean economy, etc. The
prices are relatively cheap so perhaps you can check it out. If it
looks fake to you, forget about it. Good luck!

Cheers.

The CIA also has quite the fact sheet concerning Korea...

http://www.cia.gov/cia/publications/factbook/geos/ks.html

Hope that helps!