Proof of the spectral theorem for normal operators from two lemmas

Question

I have the following lemmas that I can prove:

1. Let $T$ be a linear operator on a Hermitian space $V$ and let $W$ be a $T$-invariant subspace of $V$ . Then $W^⊥$ is $T^*$-invariant
2. Let $T$ be a normal operator on a Hermitian space $V$ , and let $v$ be an eigenvector of $T$ with eigenvalue $λ$. Then $v$ is an eigenvector of $T^∗$ with eigenvalue $\bar{λ}$.

From these two I am meant to be able to deduce the spectral theorem for normal operators, but I'm struggling. The only thing I need to finish my proof is that $V_{\lambda_1}\oplus\cdots\oplus V_{\lambda_k}=V$, or equivalently that the geometric multiplicity of any eigenvalue equals the algebraic multiplicity, or something else equivalent.

I seem to remember the proof we were given a while back following the lines of: one eigenvalue exists due to the fundamental theorem of algebra, so consider finding an orthonormal basis for $V_{\lambda_1}$, then consider $V_{\lambda_1}^⊥$, which is $T^*$ invariant, so...

But I think I might have just remembered wrong? Either way, all of the proofs that I can find on the internet use much higher level ideas, or talk about the sums of upper diagonal matrices, but I'd definitely like to (and should be able to for an exam!) be able to prove it in this way (or at least along these lines).

Sorry for any mistakes!

Edit: I have also that distinct eigenspaces are orthogonal, if that helps!