The length of pn
the length of lm

If the angles PLN = PNL, that means they are 45 each. A triangle's angles add to 180 and LPN is 90.

Knowing that these angles are equal, we also know that the length of PN is equal to LP because the angle opposite to the two sides are the same. So PN = 12.

If PN is 12, then MP = 17-12= 5.

Then use Pythagorean Theorem to find LM

5^2 + 12^2 = c^2

LM = 13