bureau, board, office, affair, conclusion, court lady, lady-in-waiting, her apartment
局 centers on a bounded space or situation with a defined role: a bureau or office, a specific affair or matter, and by extension a court lady's apartment or her position. The idea of a conclusion fits as the closing of such a bounded matter.
局 originally depicted a bent person (尸) over a mouth (口), possibly suggesting a confined space or a person speaking in a limited context. Over time it came to mean a bureau or office, a bounded affair, and related senses.
The outer 尸 looks like a flag or a bent person, and 口 is a mouth or opening. Imagine a flag marking a small office where someone speaks: that bounded space is a bureau or a court lady's apartment.
For キョク, picture a bureau that handles all the 'kyoku' (曲, tunes) for a music office: kyoku -> キョク, and the bureau manages every musical affair.
after all; in the end; ultimately; eventually
bureau; department
authorities; relevant authorities; authorities concerned
bureau director; office chief
television station
pharmacy; drugstore; chemist's (shop)
position in a game; state of the game
branch office
(playing a) game (of go, shogi, etc.)
catastrophe; cataclysm
clerk; (bureau, post-office) staff
department; bureau; section; part
locality; local area; limited area
political situation
revenue office; taxation bureau
opening (of a broadcasting station, post office, bureau, etc.); establishment
medical office (esp. in a hospital); doctor's office
the said bureau (office, department, broadcasting station, channel, etc.)
telephone exchange number
end; close; conclusion
difficult situation
general delivery; counter collection; poste restante
situation; state of things; current affairs
one game (of go, shogi, etc.)
(Japanese) Tax Bureau
budget bureau
secretariat; executive office
officials in charge
Immigration Bureau; Immigration Office
Cabinet Legislation Bureau
secretariat
Cabinet Legislation Bureau
General Affairs Bureau
post office