Analytic sets are Lebesgue measurable

Let me try to summarise the comments to give an answer:

Despite how the proofs in the references are framed, you only need to make two arguments. First you show that any Borel subset of $\mathbb{R}^2$ is the continuous image of the 'universal' Polish space $\mathbb{N}^{\mathbb{N}}$. It then follows that $\Sigma_1^1=\mathcal{A}(\Pi_1^0)$. Second, you use the completeness property of Lebesgue measurable sets to prove that they are closed under the Souslin operation. The result should then follow.

The idempotence of the Souslin operation, while interesting in its own right, is not necessary here. For anyone who's interested, it can be proved by constructing some clever bijections $\mathbb{N}^{\mathbb{N}}\times (\mathbb{N}^{\mathbb{N}})^{\mathbb{N}}\to {\mathbb{N}}^{\mathbb{N}}, \ \mathbb{N}\times\mathbb{N}\to \mathbb{N} $ and $\mathbb{N}^{<\mathbb{N}} \to \mathbb{N}^{<\mathbb{N}}$.