I would also recommend staying in the city. You'll be wasting a lot of time going to the Hyatt Incheon and then back to T2 to catch the KAL limo into Seoul. Staying at a hotel that is a stop on KAL limo is your best bet. Plaza, Lotte, Westin Chosun are all a 10 minute walk to Myungdong. Also many local eateries around those hotels. Catch the 6:30-7am KAL Limo back to T2 next am.