Superfluid ⁴He subjected to steady rotation develops a regular lattice of quantum vortices aligned with the rotation axis. We prepare this lattice in a rotating cryostat, perturb it with a constant heat flux, and visualize vortex deformation waves that propagate in the lattice and grow in energy with the forcing. Below twice the rotation rate, we show that these waves feature a continuous frequency spectrum whose structure corresponds to inertial waves. At larger frequencies, we report evidences supporting the observation of a turbulent cascade of Kelvin waves. Our experiments hence provide a direct approach to deepen our understanding of collective dynamics in perturbed quantum vortex systems across all quantum fluids.