The second “wind” is as in “I wound up the toy car and, when I released it, it zoomed all the way to the other side of the room”. In this context, a “long-winded response” is one that metaphorically winded the coils that make the speaker go.

The more primary meaning is this one (copied from Oxford Dictionary of English): move in or take a twisting or spiral course. The etymology of the verb ‘wind’ (also from ODE) is: Old English windan ‘go rapidly’, ‘twine’, of Germanic origin; related to wander and wend. Long-winded = the speaker’s words/thoughts wander in circles for a long time.