While poetry is often subjective and open to several interpretations, I think the “kayi” mentioned by Allama Iqbal is, in fact, the legendary Persian king Kay Khosrow. This is mentioned in several explanations of the Shikwa/Jawab Shikwa by several scholars.
One also has to consider that while Allama Iqbal was deeply familiar with the Persian literature, including “Shahnama” where Kay Khosrow is mentioned, we find no evidence of him making a study of the ancient origins of the Ottomans.
This does not, of course, take away from the
numerous achievements of the Turk Kayi tribe and the several illustrious rulers which came from it to whom Muslims owe a lot of debt including the Conquest of Constantinople. Just that this particular shair is not about them...