There are 2 other Frasers Properties in Seoul. Frasers Place Seoul Central and Namdaemun. Both seem closer to Gwanghwamun Square than the Insadong property, so you may want to consider these as well. On the other end of Gwanghwamun Square, there is also a Somerset Palace Seoul.

Closer to Gangnam there is an Oakwood Premier located in the COEX complex at the Gangnam area, with direct access to City Air Terminal.

These are all proper serviced apartments with a fully fitted kitchen, not sure if they would rent out for a week's stay or a min stay applies.