Harmonic oscillator with ladder operators - proof for sum rule

I’m trying verify the proof of the sum rule for the one-dimensional harmonic oscillator:
$$sum_l^infty (E_l-E_n) | langle l |p| n rangle |^2 = frac {mh^2w^2}{2} $$
The exercise explicitly says to use laddle operators and to express $p$ with
$$b=sqrt{frac {mw}{2 hbar}}-frac {ip}{sqrt{2 hbar mw}} $$
$$b^dagger =sqrt{frac {mw}{2 hbar}}+frac {ip}{sqrt{2 hbar mw}} $$

For $p$ I get $$p=i sqrt{frac{hbar}{2mw}} (b-b^dagger) $$

To solve the exercise, we need to calculate the left side. I’m still very much a novice and am not very sure how to use the ladder operators… To start, I at least tried to expand the bra-ket:
$$sum_l^infty (E_l-E_n) langle l |p| n rangle langle n |p| l rangle $$
and tried to insert the $p$ I solved:
$$sum_l^infty (E_l-E_n) (-frac{hbar}{2mw}) langle l |b-b^dagger| n rangle langle n |b-b^dagger| l rangle $$
is this correct? If yes, how do I continue? The hint says to probably use $H|nrangle=hbar(n+frac 12)|nrangle$ and I know that $H|nrangle=E|nrangle$