I have a paper giving an irreducibility test (and factoring "algorithm", in some sense of the word) for formal power series over a PID, which should eventually appear in Trans. of the AMS. In particular it applies to $K[[X]][[Y]]$.
See Theorem 1 of http://arxiv.org/abs/1107.4860
In essence: Write any polynomial $f$ in $K[[X,Y]]$ as a polynomial in $Y$ with coefficients in $K[[X]]$, and let $f_0$ and $f_1$ be the coefficients of $Y^0$ and $Y^1$, respectively. Then $f$ is irreducible in $K[[X,Y]]$ if and only if either (1) $f_0 = 0$ and $f_1$ is a unit in $R[[X]]$, or (2) $f$ has a unique factor in $K[[X]][Y]$ with constant term not a unit in $K[[X]]$. (This works, by the way, even if $f$ is in $K[[X]][Y]$.)
Added on 4/3/14: The criterion above is equivalent to the following. A polynomial $f \in K[[Y]][X]$ is irreducible in $K[[X,Y]] = K[[Y]][[X]]$ if and only if one of the following conditions holds.
(1) $f_0 = 0$ and $f_{1,0} \neq 0$.
(2) $f_0 \neq 0$ and $f_{0,0} = 0$, and if $f = gh$ with $g,h \in k[[Y]][X]$, then either $g_{0,0} \neq 0$ or $h_{0,0} \neq 0$.
Here, $f_{i,j}$ denotes the coefficient of $X^i Y^j$ in $f$, and $f_i$ denotes the coefficient of $X^i$ in $f$ as a polynomial in $K[[Y]][X]$. To test whether (2) holds or not in a given situation, you need to be able to factor polynomials over the complete DVR $K[[Y]]$. To do so you can use Hensel's lemma, Newton polygons, and other methods for factoring polynomials over local fields.