In the case of $GL_{N}$, the $L$-packets are a non-issue, and the surjective map in the local Langlands correspondence becomes a bijection. At that point, we can think of allowing the information to flow the other way. Here's a simple application.
Let $f(z)$ be a classical modular form of weight $4k+2$ for the group $\Gamma_{0}(4)$ (that is also a cusp form, in the new subspace, and is an eigenform of all the Hecke operators). If $L(f,s)$ is the $L$-function for $f(z)$, what is the sign of the functional equation for $L(f,s)$?
The sign of the functional equation is always $1$, for the following reason.
It is determined by the local components of the automorphic representation $\pi$ attached to $f$, and we only have to worry about the local components at $\infty$ (which is a discrete series representation that contributes a factor of $1$ to the sign because the weight is $\equiv 2 \pmod{4}$), and the local representation $\pi_{2}$ at $2$. The fact that the level of the modular form is $4$ shows that $\pi_{2}$ corresponds (under local Langlands) to a representation $\rho : W_{\mathbb{Q}_{2}} \to GL_{2}(\mathbb{C})$ that comes from a character $\chi$ of $W_{K}$, where $K = \mathbb{Q}_{2}(\omega)$ is the unramified quadratic extension of $\mathbb{Q}_{2}$, and that this character has order $6$. It follows that $\rho$ comes from an $S_{3}$ extension of $\mathbb{Q}_{2}$, and it turns out that there is a unique $S_{3}$ extension of $\mathbb{Q}_{2}$. From this, $\rho$ and hence $\pi_{2}$ is uniquely determined, and it turns out that the local root number of $\pi_{2}$ is also $1$.
(This fact was also observed by Atkin and Lehner in 1970, but the explanation above gives a more conceptual reason for it to be true, in my opinion.)
The Langlands correspondence for higher local fields is still at an early stage of development. I haven't really kept up with it, but here's some key points.
As the question stated, and Loren commented, the starting point is the $GL_1$ case, which is class field theory for higher local fields. Local class field theory relates the abelianized Galois group $Gal_F^{ab}$ of a local field $F$ to the multiplicative group $F^\times = K_1(F)$. For a higher local fields $E$, Kato's class field theory relates the abelianized Galois group $Gal_E^{ab}$ to the Milnor K-group $K_n(E)$.
For example, let $E = {\mathbb Q}_p((t))$. Then there's a canonical homomorphism $\Phi \colon K_2(E) \rightarrow Gal_E^{ab}$ such that for all finite abelian $L/E$, $\Phi$ induces an isomorphism from $K_2(E) / N_{L/E} K_2(L)$ to $Gal(L/E)$. This gives a bijection between finite abelian extensions of $E$ (in a fixed algebraic closure) and open, finite-index subgroups of $K_2(E)$. This is the main theorem described in
Kato, Kazuya, A generalization of local class field theory by using K-groups. I, Proc. Japan Acad., Ser. A 53, 140-143 (1977). ZBL0436.12011.
You can look at this paper to see the topology on $K_2(E)$ and more details. In particular, this suggests a possible Weil group for $E$. Namely, Kato reciprocity gives an isomorphism from a completion of $K_2(E)$ to $Gal_E^{ab}$. One might let the abelianized Weil group be the subgroup $Weil_E^{ab}$ of $Gal_E^{ab}$ corresponding to the uncompleted $K_2(E)$. And perhaps the (nonabelian) Weil group should be defined by pulling back. I.e., look at the map $\pi \colon Gal_E \rightarrow Gal_E^{ab}$, and define $Weil_E = \pi^{-1}(Weil_E^{ab})$. I haven't explored if this is the right idea though.
Kato goes beyond this, from 2-dimensional to n-dimensional local fields, and from $K_2$ to $K_n$ accordingly. These aren't hard to find, and there are surveys floating around. See the Invitation to Higher Local Fields volume, for example. Even $K_2$ is interesting, I think!
Note that Kato's paper was from 1977... so what about the Langlands program for fields like $E$? A natural first step is figuring out a suitable version of the Satake isomorphism, and the Iwahori-Hecke algebra. There's a series of papers by Kazhdan, Gaitsgory, Braverman, Patnaik, Rousseau, Gaussent (and certainly others) on the subject.
Recent landmark papers are
- Braverman, Alexander; Kazhdan, David, The spherical Hecke algebra for affine Kac-Moody groups. I, Ann. Math. (2) 174, No. 3, 1603-1642 (2011). ZBL1235.22027.
- Gaussent, Stéphane; Rousseau, Guy, Spherical Hecke algebras for Kac-Moody groups over local fields., Ann. Math. (2) 180, No. 3, 1051-1087 (2014). ZBL1315.20046.
- Braverman, Alexander; Kazhdan, David; Patnaik, Manish M., Iwahori-Hecke algebras for $p$-adic loop groups, Invent. Math. 204, No. 2, 347-442 (2016). ZBL1345.22011..
Note that a group like $SL_2(E)$ can be seen as a loop group over ${\mathbb Q}_p$. Hence the appearance of words like "loop group" and "Kac-Moody group".
The Langlands dual group certainly arises in these studies, but I haven't seen something quite as straightforward as a parameters from the Weil group (described above) to the dual group. I haven't looked too hard either, so maybe it's in there somewhere. There seems to be a fancier, more categorical, parameterization involved. I'd be tempted to bring it down to earth a bit, following Kato.
The other direction that I haven't seen -- and one that I think is worth pursuing -- is the case of (nonsplit) tori. That's important for any putative Langlands program, and should require an interesting mix of Milnor K-theory and Galois cohomology.
Best Answer
This came up in a paper of mine not so long ago, and my coauthors and I were surprised that it wasn't made explicit in the standard references, so we wrote it out ourselves:
Dembélé, Lassina; Loeffler, David; Pacetti, Ariel, Non-paritious Hilbert modular forms, Math. Z. 292, No. 1-2, 361-385 (2019). ZBL1446.11084.
See Proposition 1.3. (Strictly speaking we are describing the analogue for global fields, not local fields, but the recipe is the same.)