To fix the superscript placement and the italic m, you can use the options ScriptBaselineShifts and SingleLetterItalics in Style. For example
Plot[1 - Exp[-x], {x, 0, 3}, AxesLabel -> {"thickness (\[Mu]mμm)", Style[Row[{"power density ", Style["(",Larger], "W/m^2", Style[")", Larger]}], ScriptBaselineShifts -> {0, .5}, SingleLetterItalics -> False, "TraditionalForm"]}] 
There are some other options in Style that might help. For example if you want a smaller font for the superscript, you can use the option ScriptSizeMultipliers.