We study the hole-filling problem for the porous medium equation $u_t= \frac{1}{m} \UDelta u^m$ with $m>1$ in two space dimensions. It is well known that it admits a radially symmetric self-similar focusing solution $u=t^{2\beta-1}F(|x|t^{-\beta})$, and we establish that the self-similarity exponent $\beta$ is a monotone function of the parameter $m$. We subsequently use this information to examine in detail the stability of the radial self-similar solution. We show that it is unstable for any $m>1$ against perturbations with 2-fold symmetry. In addition, we prove that as $m$ is varied there are bifurcations from the radial solution to self-similar solutions with $k$-fold symmetry for each $k=3,4,5,\dots.$ These bifurcations are simple and occur at values $m_3>m_4>m_5> \cdots\to1$.