I have always imagined that right after the war, it could have gone either oligarchy or feudal. Given the state of affairs in KFS, I would say it is an oligarchy, the executive office being more like a board room with the five administrators being more or less equal on paper.
As for other governmental organizations, you would have the normal infrastructure ones, like urban planning, energy and water commission, natural resources department. Probably there is some form of treasury. There is definitely some sort of census bureau to keep track of status and rank in the KFS since that is important and very difficult to change your socioeconomic status. There would also be some sort of judicial branch. These guys were business leaders and contract law is big. There might be some form of social work or welfare, but I would assume that is would be really limited to "Aww, so sorry to hear you fell on bad time. I am sure we can find someone to indenture you to."
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