We consider Willmore surfaces in ${\mathbb R}^3$ with an isolated singularity of finite density at the origin. We show that locally, the surface is a union of finitely many multivalued graphs, each with a unique tangent plane at zero and with second fundamental form satisfying \[ |A(x)| \leq C_\varepsilon |x|^{-1+1/\theta_0-\varepsilon},\quad \forall \varepsilon > 0, \] where $\theta_0 \in {\mathbb N}$ is the maximal multiplicity. Examples of branched minimal surfaces show that this estimate is optimal up to the error $\varepsilon > 0$