$\begin{aligned} \mathbf{p} & =5 \mathbf{a}+2 \mathbf{b} \\ \mathbf{q} & =\mathbf{a}-3 \mathbf{b} \\ \mathbf{B D} & =\mathbf{p}+\mathbf{q}=6 \mathbf{a}-\mathbf{b} \\ \mathbf{C A} & =\mathbf{q}-\mathbf{p}=-4 \mathbf{a}-5 \mathbf{b}\end{aligned}$
$\begin{array}{r}\text { Now, }|\mathbf{B D}|=|6 \mathbf{a}-\mathbf{b}| \\ \sqrt{36 \mathbf{a}+|\mathbf{b}|^2-12 \mathbf{a} \mathbf{b}}\end{array}$
$\begin{aligned} & =\sqrt{36|\mathbf{a}|^2+|\mathbf{b}|^2-12|\mathbf{a}||\mathbf{b}| \cos 45^{\circ}} \\ & =\sqrt{36 \times 8+9-12 \times 2 \sqrt{2} \times 3 \times \frac{1}{\sqrt{2}}} \\ & =\sqrt{225}=15\end{aligned}$
and
$|\mathbf{C A}|=|-(4 \mathbf{a}+5 \mathbf{b})|$
$=\sqrt{|4 \mathbf{a}|^2+|5 \mathbf{b}|^2+40|\mathbf{a}||\mathbf{b}| \cos 45^{\circ}}$
$\begin{aligned} & =\sqrt{16 \times 8+25 \times 9+40 \times 2 \sqrt{2} \times 3 \times \frac{1}{\sqrt{2}}} \\ & =\sqrt{593}\end{aligned}$
$\therefore$ Lengths of the diagonals are 15 and $\sqrt{593}$.