Problem:

Triangle $ABC$ is right-angled at $C$, and point $D$ on $AC$ is the foot of the bisector of $\angle B$. If $AB = 6\ \mathrm{cm}$ and the area of $\triangle ABD$ is $4.5\ \mathrm{cm}^2$, what is the length, in $\mathrm{cm}$, of $CD$?