When $p \neq \ell$, if $N/K$ has Galois group $G$ then $N/K$ is tamely ramified. It follows that $N = K(\sqrt[e]{\pi}, \zeta)$ where $e$ is the ramification degree of $N/K$, $\pi$ is some uniformizer of $K$, $\zeta$ is a primitive $(p^f-1)$st root of unity and $f$ is the residue degree. It is not hard to work out $G$ given $e$, $f$ and $\pi$. From this one can work out the list of possible $G$. I'll leave the details to the reader.

The more interesting case is when $p=\ell$, which I now assume. Suppose $K/\mathbb{Q}_p$ is finite, and let $K^p$ be the maximal pro-$p$ extension of $K$ (i.e. the compositum of all finite normal $N/K$ with $(N:K)$ a $p$-power). Then $G_{K,p} := \operatorname{Gal}(K^p/K)$ is known in the literature, and can be written down in terms of generators and relations. Given this, then there is an extension of $K$ with Galois group $G$ (a $p$-group) if and only if there is a surjective homomorphism $G_{K,p} \to G$. For specific $G$, one could use a computer algebra system (like MAGMA) to find all such homomorphisms, and deduce the number of extensions with Galois group $G$.

Note that the theory behind $G_{K,p}$ uses Galois cohomology, and as such is somewhat nonconstructive. In particular, given a homomorphism $\varphi: G_{K,p} \to G$ I don't know of an easy method to produce the corresponding extension $N/K$ (i.e. the fixed field of $\ker \varphi$).

When $K$ does not contain a primitive $p$th root of unity, then $G_{K,p}$ is a free pro-$p$ group with $n+1$ generators where $n=(K:\mathbb{Q}_p)$.

When $K$ does contain a primitive $p$th root of unity, then $G_{K,p}$ is a pro-$p$ group with $n+1$ or $n+2$ generators and one relation. It is known as a *Demushkin group* and these have been fully classified, although the details are too much for a MO answer. The result of this is that given $K$, one can easily compute $G_{K,p}$. For example, $$D_{\mathbb{Q}_2,2} = \langle a, b, c : a^2 b^4 [b,c] \rangle.$$

The following articles will tell you more:

- H Koch.
*Galois theory of $p$-extensions*. (On the general theory of $p$-extensions, including the full answer in the $p \neq \ell$ case and much of the $p = \ell$ case. See chapter X in particular.)
- J. Labute.
*Classification of Demushkin groups*. PhD thesis, 1965. (Fully describes Demushkin groups in terms of generators and relations.)
- M. Yamagishi. On the number of Galois $p$-extensions of a local field.
*Proc. Amer. Math. Soc.*, 1995. (A neat summary of the previous article, and one algorithm for computing the number of extensions with a given Galois $p$-group.)