A modification of the Townsend-Perry attached eddy model is derived in order to reproduce a more realistic variation of the integral length scale. A new wavenumber range is introduced to the model at wavenumbers smaller than the Townsend-Perry k^(-1) spectrum. This necessary addition can also account for the high Reynolds number outer peak of the turbulent kinetic energy in the intermediate layer. An analytic expression is obtained for this outer peak in agreement with extremely high Reynolds number data by Hultmark et al (2012, 2013). The finding of Dallas et al (2009) that it is the eddy turnover time and not the mean flow gradient which scales with distance to the wall and skin friction velocity in the intermediate layer implies, when combined with Townsend's (1976) production-dissipation balance, that the mean flow gradient has an outer peak at the same location as the turbulent kinetic energy.