What would a hospital in King’s Landing be like and who would run it and manage the people and who would support it financially?

Interesting question!

Medieval hospitals were actually not primarily medical establishments (this didn’t happen until the rise of the Enlightenment in the 18th century and the professionalization of medicine in the mid-19th century)  – hospitals were religious institutions staffed by monks and nuns and were as much about providing housing, sustenance, and religious services for the elderly, the disabled, pilgrims, lepers, etc. – although the Rule of Saint Benedict did require monks to care for the sick. 

Needless to say, Westeros has something of a leg up with a dedicated order of Maesters who have crude anti-biotics in the form of poultices that use bread mold, crude anti-septics in the form of vinegar, and crude anesthetics in the form of milk of the poppy. So maesters would probably provide medical services to a King’s Landing hospital, if only to maintain their monopoly on the healing arts. 

However, given their other duties, I don’t think the Citadel would operate hospitals directly. More likely, you’d see the Faith operating the hospitals similar to RL medieval hospitals, and a division of labor whereby septons and septas would provide room, board, nursing, religious, and funeral services, with maesters and acolytes contracted to provide medical services in return for using the facility as a kind of teaching hospital. 

And I would imagine the doctor-to-patient ratio would be abysmal…