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.