TT correlator computation in CFT

In my work I want to find correlator $langle Delta^{{k}} |T(z)T(w)| Delta^{{l}} rangle $, where the states are some descendants of primary state $Delta$. For this, I’d like to understand how to compute correlators in operator formalism, i.e. using the commutation relations for the modes of $T(z) = sum frac{L_n}{z^{n+2}}$, but I have some difficulties.
For example, for the simplest cases we derive from OPE and Ward identities begin{equation}
langle 0 |T(z)T(w)| 0 rangle = frac{c/2}{(z-w)^4}, end{equation}

begin{equation} langle Delta |T(z)T(w)| Delta rangle = frac{c/2}{(z-w)^4} + frac{2Delta}{zw(z-w)^2} + frac{Delta^2}{z^2 w^2}. end{equation}

Now I try to reproduce the classical result via mode expansion of T(z), acting on T(w):

begin{equation} langle 0 |T(z)T(w)| 0 rangle = langle 0 |sum_n sum_m frac{L_n}{(z-w)^{n+2}} frac{L_m}{w^{m+2}}| 0 rangle end{equation}

Commutation relations $[L_n, L_m] = (n-m)L_{n+m} + frac{c}{12}(n^3-n)delta_{n,-m} $ shows, that only the term $n=-m=2$ contributes. But why all the other terms with $n=-m$ vanish?
If so, is there a way to easily obtain $langle Delta |T(z)T(w)| Delta rangle$ and more subtle correlators just from Virasoro commutation relations?

Any advice on the details of such calculations would be very helpful!