Polar form: $\vert z \vert \big(\cos\theta + i\sin\theta \big)$
$$\begin{aligned}z^2 &= \vert 12^2 + 5^2 \vert\\ z &= \vert 13 \vert \\ \arctan \frac{5}{12} &= 22.61^\circ\\ z &= |13| \big(\cos(22.61^\circ) + i\sin(22.61^\circ)\big)\end{aligned}$$
Up until now my textbook has only shown answers with $\theta$ being in radian form. Is it acceptable to write the polar form with $\theta$ in degrees like what I did above?
On
Of course it's fine; degrees and radians are just different ways of representing the same quantity. It's just that radians are more usually used due to various reasons. See this expository article on the subject. Since $|13|=13$, you can also just write the result as $$13\cos 22.61^\circ+(13\sin 22.61^\circ)i.$$
Writing the polar form using degrees instead of radians is legitimate, but not recommended. Radians are the better choice overall: one of the main reason being that they are plain numbers (https://en.wikipedia.org/wiki/Dimensionless_quantity) whereas degrees have a physical dimension.
That said, your notation is wrong. I corrected it:
$$|z|^2 = \vert 12^2 + 5^2 \vert$$ $$|z| = 13 $$ $$\theta=\arctan\left(\frac{5}{12}\right) = 22.61^\circ$$ $$z = 13 \big(\cos(22.61^\circ) + i\sin(22.61^\circ)\big)$$