In response to Minhyong's request, I am reposting my comments above as an answer:

As James Newton commented, if $L/K$ is unramified, then an irreducible $n$-dimensional
representation (over $\mathbb C$) of $Gal(L/K)$ will correspond, in the Langlands paradise,
to a cuspidal automorphic representation of $GL_n(\mathbb A_K)$.   The cuspidal automorphic representations that arise in this way are sometime (especially in the older literature)
called "Galois type".

Thus one can (more or less --- there is the issue of irreducible vs. all reps. which
I won't think about here) encode unramified extensions of $K$ whose Galois groups
admit $n$-dimensional representations in terms of Galois type cuspidal automorphic representations $\pi$ of $GL_n(\mathbb A_K)$ that are unramified at every finite prime.

Now the question arises: how many such $\pi$ are there, and can one compute them?

Being of Galois type is (conjecturally, but we are in paradise!) purely a condition
on $\pi_v$ for primes $v$ of $K$ lying over $\infty$, and in fact there are a finite
number of prescribed representations of $GL_n(K_v)$ ($= GL_n(\mathbb R)$ or
$GL_n(\mathbb C)$) which are allowed.  (E.g. for $GL_n(K)$, the possibilities are
limit of discrete series, corresponding to holomorphic weight one forms, or principal series with $\lambda = 1/4$, corresponding to Maass forms with eigenvalue of Laplacian equal to $1/4$.)  For a given $n$ and $K$, these can be enumerated.

Now since we are asking that the "weight" (i.e. the collection of $\pi_v$ for $v|\infty$) be bounded
(i.e. lie in a given finite set), and we are also asking that "level" be one (i.e. that there is no ramification at any finite prime), there are only a finite set of $\pi$
corresponding to irreducible everywhere unramified $n$-dimensional complex representations 
of $GL_n(Gal(\bar{K}/K)$.

To actually compute them (say for a fixed choice of $K$ and $n$) would be quite difficult 
(as David Hansen notes in his comment).  The reason is that the relevant $\pi_v$ for
$v|\infty$ are *never* discrete series (even when $n = 2$, and in any case, note that
$GL_n(\mathbb R)$ never has discrete series if $n > 2$, and $GL_n(\mathbb C)$ never has
discrete series when $n > 1$), and so standard applications of the trace formula to counting automorphic forms won't work.

Nevertheless, it seems that one might still be able to use the trace formula to analyze the situation, at least in principle.  For example, Selberg used his original formulation of the trace formula for $SL_2(\mathbb R)/SL_2(\mathbb Z)$ to compute cuspidal Maass forms of level 1,
and showed that the smallest eigenvalue $\lambda$ that occurs has $\lambda$ much greater than 1/4 (maybe closer to 90?).  
And we all know that it is not hard to show that there are no holomorphic weight one forms 
of level one.  So one can automorphically prove (modulo standard conjectures in the Maass form case) that there are no everywhere unramified two-dimensional complex representations of $Gal(\bar{\mathbb Q}/\mathbb Q)$.   (This is of course an incredible battle, even in paradise, for a tiny portion of the information that Minkowski gives us, but is meant just to illustrate that this approach is not *a priori* ridiculous.)

What I don't see at all from this point of view is how to study all $n$ simultaneously.
For example, one could imagine implementing this program and finding, for some $K$ and some $n$, maybe $n = 10^6$, that there are no unramified extensions $L/K$ with $L$ admitting an irrep. of dimension $\leq 10^6$.  This doesn't rule out the possibility that there is a beautiful, everywhere unramified extension $L/K$ whose Galois group's lowest degree irrep. happens to be of enormous dimension.  

The Langlands program seems to be intrinsically geared to thinking about linear representations of Galois groups, and to set the scene, you have to begin by choosing a linear group, which will then cut everything else down in a Procrustean manner.
At least superficially (and this answer reflects just superficial thoughts about the
question), it doesn't seem well adapted to questions related to the nature of
$\pi_1(\mathcal O)$, where no *a priori* linear structure is given, or indeed expected.