Simple Sketch of Mirrlees 1971
You can find a didactic exposition of Mirrlees (1971) -- including derivations of optimal tax schedule formulas for different social welfare and utility function -- in section 4.2 of "The Economics of Taxation" by Bernard Salanié (2003, First Edition).

I bet it is still somewhere in the second edition too, but I do not have a copy of it so I don't know in which section that would be.