http://www.hri.org/docs/sevres/map1.html
As I recall there was some talk about the possibility of doing so, that was what the plebescite area in the southeast of anatolia was for. If there was a Kurdish state though it would probably have been a British mandate, but perhaps not.
The best PoD I can think of is greater success for the Greeks in the Greco-Turkish war.
http://en.wikipedia.org/wiki/Greco-Turkish_War_(1919-1922)
That would keep Sevres enforced, which would mean a plebescite in the Kurdish areas. Of couse it would also mean Greece will have taken a fair bit of land from the Turks, maybe even to the point of pulling off the Megali Idea.
Turkish Armenia keeping it's independence might also have an effect of the Soviet-held portion of Armenia, creating even more butterflies. Armenian independence could potentially remove the Holocaust, as part of Hitler's reasoning was that no action was taken in response to the Ottoman massacres of Armenians in World War I. Not too likely, but it would be a nice thought.
Assuming Turkish Kurdistan goes independent in the plebescite (fairly likely I would imagine) the British would combine it with Mesopotamian Kurdistan to make a British mandate out of it. The Persian portion of Kurdistan could probably be obtained with a combination of threats and bribery by the British.
Armenia would likely maintain close ties with Greece due to shared religion and a common enemy in the Turks. Britain would also be an important ally. Relations with the Soviet Union and Turkey would be strained.
Greece would be a stronger force in the Balkans, and the monarchy would survive. With the Megali idea fulfilled Greece will be much larger in terms of both population and territory, and may move on to ambitions of dominating the rest of the Balkans and perhaps even dreams of restoring themselves to the borders of the Byzantine Empire. Conflict with the Soviet Union over the Straits will be an issue that may well drive Greece into an alliance with the Axis, as well as ambitions on Cyprus. The issue of Italian ownership of the Dodecanese might be a problem on getting Greece to join the Axis though, and perhaps Britain would gain Greek allegiance by offering a plebescite in Cyprus.
The Turkish remnant state is going to be extremely revanchist. A Soviet backed Communist regime is a good possibility, as the Turkish state will be desperately seeking a powerful benefactor to help them recover from the massive setback they have suffered.
Best way I can see for Greece to win the war is for the Liberal government to stay in power so that Greek forces are lead by competent commanders instead of political appointees. Having Kemal killed/removed from power might also be neccesary to give Greece a victory.
Of course, this ATL ends up being a lot more about Greece than Kurdistan. Kurdistan would likely end up much like the other British mandates in the Middle East, a somewhat secular regime ruled over by either a king or a dictator who bases his power on tribal ties.