Get to the platform behind the path of the second green plate. (This is the only step that we're not able to do. Alternatively get straight to the pipe or the first minion area, and change the route accordingly.)
Climb up and trigger the first minion: (video)
Kill the minion and clip into the pipe: (video up to ledgewarp), (remaining video)
Climb the pipe: (video)
Descend to the elevator room and go to the puzzle intro cutscene: (video)
Complete the puzzle: (video)
Clip into the pipe and climb to the top: (video)
Clip through the gate and complete the level: (video)