We derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring $\mathfrak{o}$. To this end, we develop Hecke-theoretic techniques for the enumeration, by two distinct invariants, of sublattices of an $\mathfrak{o}$-lattice of finite rank endowed with a non-degenerate symplectic form.