The Model K is the predecessor to the Model AW, but also was produced alongside the AW for a brief period. Gear ratio should be basically the same as the AW.
There are major differences in the K series of hubs, depending on whether the hub is from before or after 1933. It is noteworthy here that the K underwent several changes to the parts over the course of its life. A 1933-37 K hub (and especially 1936-37) has quite a bit common with the early AW, whereas the pre-1933 K hub is much more different from the AW internally.
These changes were often made to simplify and cheapen production of the hubs. The AW is a bit simpler and was apparently cheaper to produce than the K, though the later K hubs shared more features in common with the AW than the earlier K hubs. The AW (and maybe the very late K, to the extent AW parts may work) is MUCH easier to get replacement parts for today. The K and the AW were also produced at the same time for a brief period in 1936-37.
I encourage you to read Tony Hadland's fact sheet on the K. Hadland is a wealth of knowledge about all things Sturmey (and Raleigh). He has written several good books on these subjects.
https://hadland.files.wordpress.com/2012/07/k-series-s-a-hubs.pdf
In the U.S., I would rate the rarity as:
pre-1933 K model: rare to find in the U.S.
1933-37 K model: uncommon but can be found with some looking/patience.
Early AW: not very common, but turn up several times per year on eBay and similar. More common than K.
Post-war AW: common in the U.S.