If $E/F$ is a Galois extension, $[E:F] = 10$, and the intermediate fields $K,L$ are both degree 2 over $F$, then $K \cong L$.
Using the tower formula, we know $10 = [E:F] = [E:K][K:F] = n \cdot 2$ which means that $[E:K] = 5 = [E:L]$. Since $K,L$ are intermediate fields, they must be Galois extensions. By the Fundamental Theorem of Galois Theory, $|\text{Gal}(E/K)| = 5 =|\text{Gal}(E/L)|$ , so both $\text{Gal}(E/K)$ and $\text{Gal}(E/L)$ are cyclic, isomorphic to $\mathbb Z / 5 \mathbb Z$ , and hence normal subgroups. But
$\text{Gal}(E/K) \cong \text{Gal}(E/L)$ , so I'm thinking this is enough to conclude that the intermediate fields are isomorphic, but I don't know for sure since I can't find anything that says isomorphic Galois groups correspond to isomorphic fixed fields. Thanks.