There's a series that started airing where I live, 謎解きはディナーのあとで, and along with greatly enjoying the style, I've been very fascinated by the butler's very formal manner of speaking. What I never got, though, was what powers お忘れなきよう, which is what he says after telling his mistress about what the main lesson she needs to take with her is. Does anyone know? Is it something along the lines of "Let us never speak of this again"?
First, it is なきよう, not なきょう.
なき is a literary form of ない, and よう is the same as ように, which is often used at the end of a polite request.
お忘れなきよう ＝ お忘れのないようになさってください
In case you are not familiar with なさってください, it is a politer form of してください.
Therefore, the phrase means "Please do not forget."