A week with Seoul and Jeju has a lovely change of scale: palace courtyards and tiled-roof lanes first, then a volcanic crater, open sea and waterfall paths. Three nights in each place give the trip two full sightseeing days in Seoul and two on the island, with an arrival day, a flight day and departure around them.
The island stay uses Jeju City as a single base. That keeps the airport and Dongmun Market within reach while the car opens up a different coast each day. Seongsan lies to the east; Seogwipo and Jungmun make a separate southern outing. These are days out from the city, rather than short hops between neighbouring sights.
Gimpo is the Seoul departure airport for this plan. A morning flight leaves part of the afternoon for the market and a first island dinner, once the rental-car handover is finished. An international flight from Jeju keeps the final day simple. Returning through Incheon adds a domestic flight and a change of airports.