Quick guess might be the window regulators are going/gone bad. The cables stretch over time and the windows won't drop properly. From experience, spring for the OEM regulators, if you R&R them. Have you tried tripping the micro switch with your finger or pencil? Does that turn the light off? Maybe there is a micro switch alignment issue?
I would be curious why both cable ends are broken. Be careful with the top transmissions. As I understand, there is a very specific procedure for getting those setup correctly. I have not done it myself. Pelican Parts has an in depth DIY. Maybe you can glean something from it?
www.pelicanparts.com/techarticles/Boxster_Tech/BODY_Convertible_Top_Gear_Replacement/BODY_Convertible_Top_Gear_Replacement.htm