This isn't my first attempt at this. Several years ago I tore apart two solar "tiki torch" lights. As can be seen in the pictures I mounted the solar panels on the damper and ran wires down the flue to where I mounted the lights. They worked. Sort of. They really didn't put out much light. You could tell that they were on after dark when most of the room lights were out. Their flicker was quite nice, but just not bright enough. I should have added more.
The lower part of the flue has been boarded up. A long 2x2 is screwed into that piece of wood and from there you can see the lights hanging. I pretty much left this as it is. All that changed is the light on the end and the wiring. The damper doesn't seal well. I may need to block it off with some extruded foam to keep cool winter drafts out and to keep the a/c from bringing in outside humid air.
Another benefit of doing this project is that this is where my Mrs. sets up one of those little winter village scenes during the holidays. This solar light will be unhooked and I'll attach it to some 12v LED christmas lights to light up the scene instead.