The Foxes moved up to second in the Championship in January but have won just twice in 16 league games since February to drop down to eighth place.
Leicester must beat Nottingham Forest at the City Ground on the final day of the season to have any chance of claiming a playoff place.
"It's about the team being promoted," King told BBC Radio Leicester.
"If we have not done that then we cannot come away and say the season has been a success. The owners want promotion, but no one wants promotion more here than the players. It is a massive game for the club, owners and fans as well.
"It is down to us to make sure that we go out there and perform and hope we come away with the three points."
Leicester are two points adrift of sixth-placed Bolton Wanderers.