If I flip the Kiln and target, lets say, a 1-drop my opponent controls and I also stun their 5-drop, can I then play Cosmic Order during the recovery phase to force them into KO-ing their 5-drop? Or are they allowed to choose their 1-drop to KO, and not KO it since Kiln says they cannot KO it...?

They may choose to KO their 1-drop. Since it can't be KO'd, you'll be down one character. This is akin to dealing with the Anti-Monitor vs. a Lazarus Pit'd character. You must attempt to KO something. That it can't be KO'd is irrelevant.