jmc

Since $z^2=156+65i$, we must have $|z^2|=|156+65i|=|13(12+5i)|=13|12+5i|=13(13)=169$. We also have $|z{|}^2=|z|\cdot |z|=|(z)(z)|=|z^2|$, so $|z^2|=169$ means that $|z{|}^2=169$, which gives us $|z|=\sqrt{169}=\boxed{13}$.