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The original reference is Kerr, R.P. Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237 (1963).
you can also read this paper , it is very useful: A Possible Intuitive Derivation of the Kerr Metric in Orthogonal Form Based On Ellipsoidal Metric Ansatz.
I'm surprised no one mentioned Chapter 6 of Chandrasekhar's Mathematical Theory of Black Holes. Then again, I personally found his derivation so opaque that I wrote my own de novo derivation in Boyer-Lindquist coordinates using the Cartan formalism.
I think if we follow the Ray D' Inverno book intitled Introducing Einstein Relativity then we get easily this metric. on chapter 19 it is explanied in a very simple method
Here is a review of the process of discovery by the man himself: icra.it/icranet_report/08Kerr.pdficra.it/icranet_report/08Kerr.pdf The OP might like to read this and see if he still has the fortitude for the task!