[DByers]

The black box inside is the xfmr. You are looking at a throw away unit. Any home improvement store will have something like that dirt cheap (th whole fixture). By the time you waste your time trying to figure out whats wrong and buying replacement parts you could have been done and on with life. If your testing 11v to 12v is what you should have on the load side of the xfmr and 115v to 120v on the line side. Lamp socket should be 11v to 12v also.

Black and white form the flex into the suface mounted junction box is your hot and nuetral wire. Green is your ground. Is this switched/dimmed from the unit or from a wall box somewhere in the kitchen? If its on a dimmer located on the wall then that may be your problem as those electronic xfmrs are not designed to be dimmed with a standard dimmer. Will shorten the life of the xfmr.

Hope that helps a little bit