I tried that and it works, but I found a new way to flummox that puzzle up. I left the robot at the top of the stairway, so there was no one to keep the door to the center chamber open. I still think that if the little window at the top of the center chamber were just a bit lower, that would solve the whole problem.
Also I would never have thought of jumping down from between the two bridges at the top of the tower. At that height, she really should have taken some kind of fall damage. In real life, any fall from more than three times your height causes traumatic injury, so unless she's wearing magic "Aperture Science long-fall sandals" I would think she would break her ankles and have to have the robot carry her around for the rest of the game.