Finding all invariant subspaces of $T_A:\mathbb{C}^n \to \mathbb{C}^n$ where $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)$

Question

Let $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)\in \text{Mat}_n(\mathbb{C})$ when $\lambda_i\neq\lambda_j$ for every $i\neq j$. I need to describe all the $T$-invariant subspaces of $T=T_A:\mathbb{C}^n \to \mathbb{C}^n$.

$T_A$ stands for $T_A(v)=Av$.

I proved two statements that I believe are necessary to solve this problem:

1. If $T:V\to V$ is nilpotent shift operator over some basis $\mathcal{B}=(v_1,...,v_n)$: $\forall v_i\in \mathcal{B}, T(v_i)=v_{i-1}\ (e_0:=0)$, then every $T$-invariant subspace $W\leq V$ is of the form $W=\text{Span}(v_1,...,v_k)$ for some $0\leq k\leq n$.
2. Let $T:V\to V$ operator and let $\lambda \in\mathbb{F}$. Then $W\leq V$ is $T$-invariant subspace if and only if it is $T+\lambda I$-invariant.

So I understand that when given some $T_A$-invariant $W\leq V$ subspace, I can show that it is of the form $W=\tilde{W}_{\lambda_1}\oplus...\oplus \tilde{W}_{\lambda_l}$ where $\lambda_1,...,\lambda_l$ are eigenvalues of $T_A$ and $\tilde{W}_{\lambda_i}=\ker(T-\lambda_i)^{n_i}$. Then I want to say that each $\tilde{W}_{\lambda_i}$ is of the form $\text{Span}(v_1,...,v_k)$ due to statments $1+2$.

I'm having a hard time formalizing this idea and finding the connection between the Jordan blocks to the invariant subspaces.