The KE lounges in the new terminal (which DL also uses) are pretty bad.

AFAIK the cultural center is on the top floor of the other old terminal (T1) and I'm not sure you can go there with a KE boarding pass. I think this area is airside but I'm not sure.

My instinct would be to try for one of the free tours offered by the airport. Given your timing, I wouldn't try to go downtown just to shop for cosmetics, but you should check prices and selection at the airport dutyfree. Alternatively, you could try to explore Incheon City or get a day pass (I assume this is still available, but check) for the spa and health club, including indoor pool(s) at the Grand Hyatt, which has frequent shuttle service to/from both terminals.