If I know the index $(\mathbb{Q}_p^{\times} : (\mathbb{Q}_p^{\times})^n)$ for some $n \in \mathbb{N}$, is it possible to know how many field extensions of $\mathbb{Q}_p$ of degree $n$ there are?
This seems to work for $n=2$. I'm not sure in general. Thanks for your help.
That index is very easy to calculate, since $\mathbf Q_p^\times \cong \mathbf Z \times \mathbf Z_p^\times \cong \mathbf Z \times \mathbf F_p^\times \times \mathbf Z_p$. So you might as well be asking how many extensions of a given degree $\mathbf Q_p$ has, without any assumptions on what is known.
The answer is, in general, quite difficult. I don't know of any simple formula. If you're willing to count extensions with weights other than $1$, there is a very nice mass formula of Serre which you can find by searching the literature.
Abelian extensions of $\mathbf Q_p$ are easier to count, by using the Artin reciprocity isomorphism of local class-field theory. This is why it works well for $n=2$. You might want to start there.
Otherwise, the question is much more difficult. The absolute Galois group of $\mathbf Q_p$ is quite a beast (though much less of a beast than the Galois group of $\mathbf Q$).