Computing Hodge Laplacian

Question

I have a confusion regarding computation of the Hodge Laplacian on an $m$-dimensional manifold. Conisder $$\Delta f = d^{\dagger}d f = -*d*(\partial_{\mu}f ~dx^{\mu})\\=-*d\left(\frac{\sqrt{g}}{(m-1)!}\partial_{\mu}f g^{\mu\lambda}~\epsilon_{\lambda \nu_2\cdots \nu_m}dx^{\nu_2}\wedge\cdots\wedge dx^{\nu_m}\right)$$ Now, I am kind of stuck here because $\epsilon_{\lambda\nu_1\cdots\nu_m} = g_{\lambda\alpha_1}g_{\nu_2\alpha_2}\cdots g_{\nu_m{\alpha_m}}\epsilon^{\alpha_1\alpha_2\cdots\alpha_m}$ which means the Levi-Civita tensor is implicitly a function of the metric. Now, how does one proceed from here and show that $\Delta f = \frac{1}{\sqrt{g}}\partial_{\mu}(\sqrt{g}g^{\mu\nu}\partial_{\nu})f$ ? Please help.