I haven’t taken the turn signal stalk apart before, but with the multi-function setup—courtesy light blink, wiper, turn signal, and washer—there’s a lot happening inside, so it’s possible there’s an internal issue. However, are you sure you’re pressing the correct button? On the stalk, there’s a button on the end that flashes the parking lights, and the clicking sound is likely a relay in the electrical box beneath the window. The windshield washer actuator is actually the inner ring, which needs to be pushed to activate the washer function. The inner ring is more like a shoulder you press, rather than a button at the end of the stalk.