When I was nearing completion of my PhD at Princeton, my advisor asked me to cover for his class (for which I was the TA) when he went off to a conference. The class (MAT451) was a senior-level undergraduate topics class, and the topic for that semester was an introduction to Lorentzian geometry and general relativity. For my lecture (or two), I talked about how one might have discovered the Kerr metric, starting from some knowledge of the Schwarzschild black hole.
Much of the mathematics that is used or that motivates the discussion in this set of notes actually postdates the discover of the Kerr solution in the early 1960s. So take this discussion as a romantics' imagining of what could have happened.