If the detailer displayed gross negligence and pushed down on the open window while it was cracked to clean the edge of the glass (which inturn caused one tooth to break) then maybe, but to see that the whole mechanism was worn out and multiple teeth were in various states of disrepair, there is no reason he should have to be responsible.
But this is the reason why a detailer should not open a window to clean the edge. Its not because they are being lazy, its because the good detailer knows that a customer may pull this kind of stunt. Having worked in an independent repair shop, unless opening the window was critical for the repair, we had a policy of not opening a customers window for this very reason.
Next time, to protect yourself, have a form that you and the customer use upon vehicle arrival. Have it state that you are not responsible for the vehicle's mechanicals while in your possession. Also inspect the vehicle when it arrives and have the customer actuate the windows to ensure that they all indeed work. If they refuse to do so, you refuse to roll them down for complete cleaning.