If your ticket was issued by DL, KE will not provide a hotel room, even if the flight is on KE metal.

Options are to pay for an off airport hotel in Inchon, rent an airside room at the the airport's hotel or waiting in the airport where there is free wifi, lots of restaurants and some quiet areas with shower access. I'd recommend leaving the airport and staying at the Hyatt Inchon. There is a direct shuttle bus to the hotel, nice rooms, a fine restaurant & bakery and a health club with a swimming pool. A stay there is a nice way to break up a long flight.