I think an even easier answer (although it should more or less boil down to Ekedahl's) is to apply snake lemma to the following diagram - noting that the middle vertical arrow is an isomorphism
$$
0\to H^2(K_{nr}/K,A_{nr}^\times)\to H^2(K_{nr}/K,K_{nr}^\times)\to H^2(K_{nr}/K,Q)\to 0
$$
$$
0\to H^2(\bar{K}/K,A_\bar{K}^\times)\to H^2(\bar{K}/K,\bar{K}^\times)\to H^2(\bar{K}/K,Q)
$$
where I apologize for not putting the vertical arrows but I do not how to write diagrams: they are inflation maps (they go down). These sequences are the cohomology sequences coming from the exact sequence
$$
1\rightarrow A_\bar{K}^\times \rightarrow \bar{K}^\times\rightarrow Q\rightarrow 1
$$
given by taking $p$-adic valuation. I write $Q$ instead of $\mathbb{Q}$ because the latter is not discrete, so profinite Galois cohomology must be taken with a grain of salt: simply define $Q=\lim \frac{1}{e}\mathbb{Z}$ and filter $\mathrm{Gal}(\bar{K}/K)$ by subgroups fixing an extension of $K$ with absolute ramification index $e$. Then you can apply Proposition 8 in Serre's "Cohomologie Galoisienne". The point is that the inductive limit topology makes any $\frac{1}{e}\mathbb{Z}\subset\mathbb{Q}$ open (and the limit is indeed discrete), but this is not the case with the usual euclidean topology of $\mathbb{Q}$.

Of course, in order to apply Snake Lemma, one needs to check some facts:

1) The three zero showing up are really $0$

2) The right vertical arrow is injective.

For what concerns the zero on the right of the first sequence, it is just cohomological triviality of units in unramified extension (going to the limit): you find it in Serre's paper in Cassels and Fröhlich, Proposition 1 (it is also referred to in Ekedahl's answer).
For what concerns the two zeros on the right AND injectivity of the right vertical arrow, just apply inf-res sequence to
$$
1\to\mathrm{Gal}(\bar{K}/K_{nr})\to \mathrm{Gal}(\bar{K}/K)\to \mathrm{Gal}(K_{nr}/K)\to 1
$$
for the module $Q$ showing up above.