Brake lights hesitate to come on

Been having this problem for a while. Never could find where to fix this issue. I push the brake pedal and the lights don't come on until almost the end of the travel. I went to Toyota and didn't want to pay them hundreds of dollars to fix a simple thing. I asked what to do and they told me. However, I cannot figure out what to do lol. Does anyone know how to fix this?

 

Possibly your brake light switch needs adjusted, it's on the brake petal up under the dash.

 

That's what I attempted to do. How exactly do you do that? I loosened the nut behind the fixed piece. Couldn't really move anything else.

 

Disconnect the plug and wire attached to it, and unscrew it several turns. Tighten nut, plug in the plug.