Nanomechanical spectroscopy (NMS) is a recently developed approach to determine optical absorption spectra of nanoscale materials via mechanical measurements. It is based on measuring changes in the resonance frequency of a membrane resonator vs. the photon energy of incoming light. This method is a direct measurement of absorption, which has practical advantages compared to common optical spectroscopy approaches. In the case of two-dimensional (2D) materials, NMS overcomes limitations inherent to conventional optical methods, such as the complications associated with measurements at high magnetic fields and low temperatures. In this work, we develop a protocol for NMS of 2D materials that yields two orders of magnitude improved sensitivity compared to previous approaches, while being simpler to use. To this end, we use mechanical sample actuation, which simplifies the experiment and provides a reliable calibration for greater accuracy. Additionally, the use of low-stress silicon nitride membranes as our substrate reduces the noise-equivalent power to $\textrm {NEP} = 890$ fW $\sqrt{\mathrm{Hz}}^{-1}$, comparable to commercial semiconductor photodetectors. We use our approach to spectroscopically characterize a 2D transition metal dichalcogenide (WS2), a layered magnetic semiconductor (CrPS4), and a plasmonic super-crystal consisting of gold nanoparticles.