Strong gravitational lensing has been identified as a promising astrophysical probe to study the particle nature of dark matter. In this paper we present a detailed study of the power spectrum of the projected mass density (convergence) field of substructure in a Milky Way-sized halo. This power spectrum has been suggested as a key observable that can be extracted from strongly-lensed images and yield important clues about the matter distribution within the lens galaxy. We use two different N-body simulations from the ETHOS framework: one with cold dark matter and another with self-interacting dark matter and a cutoff in the initial power spectrum. Despite earlier works that identified k≳100  kpc[superscript -1] as the most promising scales to learn about the particle nature of dark matter we find that even at lower wave numbers—which are actually within reach of observations in the near future—we can gain important information about dark matter. Comparing the amplitude and slope of the power spectrum on scales 0.1≲k/kpc[subscript -1]≲10 from lenses at different redshifts can help us distinguish between cold dark matter and other exotic dark matter scenarios that alter the abundance and central densities of subhalos. Furthermore, by considering the contribution of different mass bins to the power spectrum we find that subhalos in the mass range 10[superscript 7]-10[superscript 8]  M[subscript ⊙] are on average the largest contributors to the power spectrum signal on scales 2≲k/kpc[superscript -1]≲15, despite the numerous subhalos with masses >10[superscript 8]  M[subscript ⊙] in a typical lens galaxy. Finally, by comparing the power spectra obtained from the subhalo catalogs to those from the particle data in the simulation snapshots we find that the seemingly-too-simple halo model is in fact a fairly good approximation to the much more complex array of substructure in the lens.Alfred P. Sloan FoundationUnited States. National Aeronautics and Space Administration (ATP Grant NNX17AG29G)Massachusetts Institute of Technology (RSC Award)Kavli Foundation