Prove the Weyl's complete reducibility Theorem on finite-dimensional $\mathfrak{g}-modules$ by Kostant's $\mathfrak{n}$-cohomology result

Question

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)).

But I have no idea about its proof.

Any suggestion will be very appreciated. Thank you very much.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $\mathfrak{n}$ be the usual upper nilradical and $\lambda$ be a dominant integral weight. Then the Kostant's $\mathfrak{n}$-cohomology result can be stated as follows:

$\text{H}^i(\mathfrak{n}, L(\lambda)) =\bigoplus_{w\in W,\text{ } \ell(w)=i} \mathbb{C}_{w(\lambda+\rho)-\rho}$

Question (1): How to use the Hochschild-Serre Spectral sequence to prove Kostant's $\mathfrak{n}$-cohomology result.

Question (2): How to derive the Weyl's Theorem from Kostant's $\mathfrak{n}$-cohomology result.

Thanks very much!