Kay is how you spell the letter k. I was told it was a short form for the Latin phrase 'omnis korrectas'
Meaning 'all correct'. However it is also a Greek short form for 'all is fine/well'. And it was adopted as a telegraph signal to indicate an open line, where we get the meaning, "I am listening to you". It was used as a call sign and hand signal for pilots and ground agents in ww1 to indicate " you are good to go," meaning the chucks were off the plane...which is where we get its meaning time to start. Also it was a military signal used as far back as the american civil war, indicating after a battle zero kills or fatalities. Where we get the meaning, I'm surviving, or I am still here. Used as a shipment symbol, it means this is acceptable or passes customs or this will do.
From the scotch Irish entomology of och aye, is the more enthusiastic oh, yes! Roughly, "with pleasure".
And of course there are variations of this in root form in most of your European languages, because of their familial origins.
As well as certain native american languages.
As such it is a discovered universal word.
But back to the question of the correctness of OK, I was told in formal writing you write it out, most especially if it is used as an affirmative.
But for Latin for your characters? In your story's time wouldn't it be rather SVV? Si Vales Valeo? If you are well I am well?
Or the more protracted, hi! Heus! Greetings good morning or evening or welcome! Salve! How are you? Quid ageis? I am fine! Valeo! And you? Et tu? Good/so so... Bene /admodem bene.