Consider the following square: $\require{AMScd}$
\begin{CD}
pt \sqcup pt @>>> \mathbb{A}^1_k\sqcup \mathbb{A}^1_k\\
@V  V V @VV  V\\
pt @>>> \text{Spec}(k[x,y]/(xy))
\end{CD}
This is a pullback square. The two affine lines are getting mapped to the two intersecting lines in the obvious manner. The fiber over the intersection point is just two disjoint points. This square is also a blow-up square since the right vertical map induces an isomorphism on the complement of the points. Such squares induce long exact sequence in the following manner:
$$\ldots \rightarrow G_n(pt\sqcup pt)\rightarrow G_n(pt)\oplus G_n(\mathbb{A}^1_k\sqcup \mathbb{A}^1_k)\rightarrow \text{Spec}(A)\rightarrow \ldots$$
Looking at this it is easy to see that since $G_n(pt\sqcup pt) \rightarrow G_n(\mathbb{A}^1_k\sqcup \mathbb{A}^1_k)$ is an isomorphism the map $G_n(pt\sqcup pt)\rightarrow G_n(pt)\oplus G_n(\mathbb{A}^1_k\sqcup \mathbb{A}^1_k)$ is a split injection. Now you can see that $G_n(A)\cong G_n(pt)$.