$\newcommand{\C}{\mathbb C}$I think this is OK. The first step is the inclusion of $\C[X,Y]$ into its fraction field which is $\C(X,Y)$. For each irreducible polynomial $f$ (normalised so that the top degree monomial for some ordering is $1$) we map $\C(X,Y)$ to $\C(X,Y)/\C[X,Y]_{(f)}$ and then we map $\C(X,Y)\rightarrow\bigoplus_f\C(X,Y))/\C[X,Y]_{(f)}$ which is the next step in an injective resolution, the kernel of this map is clearly $\C[X,Y]$. Finally, the cokernel of this map is injective (as the global dimension of $\C[X,Y]$ is $2$).
Addendum: A systematic way of getting this resolution as well as identifying the last term is to note that the Cousin complex of $\C[X,Y]$ is an injective resolution (Hartshorne: Residues and duality, SLN 20, p. 239) which in degree $p$ is the sum of the injective hulls of the residue fields of points of dimension $p$.
$\newcommand{\C}{\mathbb C}$I think this is OK. The first step is the inclusion of $\C[X,Y]$ into its fraction field which is $\C(X,Y)$. For each irreducible polynomial $f$ (normalised so that the top degree monomial for some ordering is $1$) we map $\C(X,Y)$ to $\C(X,Y)/\C[X,Y]_{(f)}$ and then we map $\C(X,Y)\rightarrow\bigoplus_f\C(X,Y))/\C[X,Y]_{(f)}$ which is the next step in an injective resolution, the kernel of this map is clearly $\C[X,Y]$. Finally, the cokernel of this map is injective (as the global dimension of $\C[X,Y]$ is $2$).