I do not know how to fix your problem but I will give you some information that 'might' be helpful if you take it to the dealer.
The light on my drivers side door handle quit working in either October or November. When I took my vehicle to the dealer for scheduled maintenance I told them about the light on the handle. They said that they would need to order a new one. That was in early November. Fast forward to the 2nd week of January. After multiple calls to the parts department and GM customer service I still did not have a new door handle.
According to GM they don't have any. How is this possible I asked as they are still building Malibu's that have door handles in them? Apparently GM changed parts suppliers and is backordered accross the country. GM did infact change the part number for this item in early December, showing the change. So after over two months of waiting...no new door handle, no pretty blue light.
Fast forward to yesterday...that is a little confusing...anyways, I just had my car to the same dealer to get my bowties colormatched and I asked if they could once again look into the door handle issue with GM.
Apparenly someone from the body shop (because of the painting I was having done) decided to look at it for himself, insead of the tech from the service department. He found that a stupid plastic connector had broke causing a short in the connection. They replaced the connector and walla...light glorious light.
I would feel free to tell a rep that the connector to the light is damaged or shorted out so you don't have the same issue as me.
If your handy, you could probably pop out the handle and solder the connection...but it should still be under warranty...ha free service.