About prior to 1900: According to the two maps in the links below, the major Kurdish areas were outside effective Ottoman control in the first half of the 19th centuries. Unfortunately, I do not have much knowledge about the details here. Maybe someone else could assess the possibility of establishing a Kurdish state some time between 1800 and 1840? Would it be able to survive? Or would the Ottomans be able to reconquer the area?
http://gulf2000.columbia.edu/images/maps/Islamic_States_1800_lg.png
http://gulf2000.columbia.edu/images/maps/Islamic_States_1840_lg.png
Just checking my sources it seems like there were a couple reasons why a Kurdish state in that time period is improbable:
Firstly demographics isn't on their side, the city of Dyabakir the capital of eastern Anatolia lost over half their population and the mountainous village areas were mainly immigrating to the Coastal cities like Izmir, Istanbul and Aleppo which were growing.
The Ottomans were embroiled in the Oriental crises already, and it was at this time that the dynasty was being propped up by Russian, Austrian and British armies. A potential Kurdish leader following along the example of Mehmet Ali would probably wind up being put down by Russian troops, especially if any Christians are harmed by this Kurdish leader.