Hi Noel,
At the moment, I'm thinking your Kay is a post WWII model. Although the headstock style on your instrument was introduced in 1938, I don't think it has a 1938 logo. Judging from the photos, your archtop is made completely of Honduras mahogany (If I'm wrong about this, let me know).
Here is a picture of an all mahogany K20 from 1939. Although it is difficult to make out, I don't think the logo looks like the logo on your guitar. The K20 also has 9 pearl dots rather than the 5 dots seen on your guitar. It is possible you have the K5 which is a smaller version of the K20 but I don't have a picture of the K5 to confirm that possibility.
Now back to my hypothesis that your Kay is post WWII. Here is a picture of a 1947 K36. It is hard to make out, but I think the logo matches your logo. However, notice this K36 also has the 9 dots rather than the 5 dots on your instrument. Other than that, I think we have a decent match.
The problem with assigning an exact date is that these guitars were made from 1947 to 1951. So, I don't think we have an exact model, or an exact date but if the guitar were mine, I would focus on trying to find 47-51 Kay literature to establish a positive ID.