I am sorry if I see this question only now, but since no one gave the following answer, it seems worth posting it.
There is a general formula for the number of extensions of degree $d$ of a $p$-adic field $K$ contained inside a fixed algebraic closure $\overline{K}$, which is given by $$ \# \{ L \mid K \subseteq L \subseteq \overline{K}, \, [L \colon K] = d \} = \sigma(h) \cdot \sum_{j = 0}^m \frac{p^{m+j+1} - p^{2j}}{p - 1} \cdot (p^{\varepsilon_p(j) \cdot d \cdot d_0} - p^{\varepsilon_p(j - 1) \cdot d \cdot d_0}) $$ where:
- $\sigma$ denotes the sum of divisors function;
- $h, m \in \mathbb{N}$ are the unique natural numbers such that $p \nmid h$ and $d = h \cdot p^m$;
- $\varepsilon_p(j) := \sum_{k=1}^j p^{-k}$ if $j \geq 1$, $\varepsilon_p(0) := 0$ and $\varepsilon_p(-1) := -\infty$, i.e. $p^{\varepsilon_p(-1) \cdot d} = 0$. In particular, $p^{\varepsilon_p(j) \cdot d} \in \mathbb{N}$ if $-1 \leq j \leq m$;
- $d_0 := [K \colon \mathbb{Q}_p]$.
This formula is due to Krasner, and has been proved in the paper "Nombre des extensions d'un degré donné d'un corps $\mathfrak{p}$-adique". The proof uses the same analytic techniques that go into the proof of the (much more famous) Krasner lemma.
Observe that this number is different from the number of $K$-isomorphism classes of extensions of $K$ having a given degree. This is of course due to the presence of non-Galois extensions. Here are two examples of this phenomenon:
- if $q \neq p$ is a prime then there are $q + 1$ fields $K \subseteq L \subseteq \overline{K}$ having degree $[L \colon K] = q$, but there are only two isomorphism classes of these fields: one containing the only unramified extension, and the other containing the tamely and totally ramified extensions;
- if $p \geq 3$ there are $1 + p + (p^2 - p) \cdot p$ extensions $\mathbb{Q}_p \subseteq L \subseteq \overline{\mathbb{Q}_p}$ such that $[L \colon \mathbb{Q}_p] = p$, but they form $1 + p + p^2 - p = p^2 + 1$ isomorphism classes. $p + 1$ of these contain a unique extension (which is Galois over $\mathbb{Q}_p$) and every other isomorphism class contains $p$ extensions (see for example Proposition 2.3.1 in the paper "A database of local fields" by Jones and Roberts).
Finally, let me remark that this formula is related to Serre's "mass formula", which is valid in any characteristic. This formula says that a certain "count" of totally ramified extensions of a local, non-Archimedean field $K$ of degree $d$ is equal to $d$. More precisely, $$ \sum_{L \in \Sigma_d} (\# \kappa)^{d - 1 - \mathrm{v}_K(\mathrm{disc}(L/K))} = d $$ where $\Sigma_d$ denotes the set of totally ramified extensions of $K$ which have degree $d$, and $\kappa$ is the residue field of $K$. Observe that if $p \nmid d$ then the formula can be written simply as $\# \Sigma_d = d$. Two useful references for this are:
- Serre's original paper "Une 'formule de masse' pour les extensions totalement ramifiées de degré donné d'un corps local";
- Krasner's paper "Remarques au sujet d'une note de J.-P. Serre...", in which he reproves the formula using his methods.