jmc

Time to draw! To find the perimeter, we need $AB$ and $AC.$ We recognize $△ABK$ as a $3:4:5$ triangle and and $△ACK$ as a $45^{\circ }-45^{\circ }-90^{\circ }$ triangle, which means that $AB=10$ and $AC=6\sqrt{2}.$ As for $BC,$ we have $BC=BK+CK=14.$ Therefore, our answer is $AB+AC+BC=\boxed{24+6\sqrt{2}}.$