Scalings of the streamwise velocity energy spectra in turbulent boundary layers were considered in Part 1 (Baars & Marusic, J. Fluid Mech., vol. 882, 2020, A25). A spectral decomposition analysis provided a means to separate out attached and non-attached eddy contributions and was used to generate three spectral sub-components, one of which is a close representation of the spectral signature induced by self-similar, wall-attached turbulence. Since sub-components of the streamwise turbulence intensity follow from an integration of the velocity energy spectra, we here focus on the scaling of the former. Wall-normal profiles and Reynolds-number trends of the three individual, additive sub-components of the streamwise turbulence intensity are examined. Based on universal trends across all Reynolds numbers considered, some evidence is given for a Townsend-Perry constant of, which would describe the wall-normal logarithmic decay of the turbulence intensity per Townsend's attached-eddy hypothesis. It is also demonstrated how this constant can be consistent with the Reynolds-number increase of the streamwise turbulence intensity in the near-wall region.