Okay, here it is. I took the handle with me to work and I find a very nice fix for it. I don't have my car here so the final verdict will have to be made over the weekend but it looks and feels very sturdy. Here is what I did in case anyone finds it useful:
Looking at the design of the handle I realised that I had to find something to stick in the holes. The problem was that what I put in the holes had to be a very snug fit that would not push all the way through. If it did the handle would come loose again. My initial thought was to ask my technician to turn a couple of plugs made of hard plastic. It looked like the holes might have a bit of taper to them so the plugs would stick in nicely. If not, they would have to be glued somehow but that would be a less durable option. However, when I came into my office I found a piece of high pressure oxygen hose that I had brought in from my laboratory and it looked like the diameter might be suitable. The hose was a bit bigger but it has a bit of give to it and I was sure it would fit. The benefit of using the hose is that it can be forced in and as it expands it makes for the perfect fit. The hose has thick reinforced walls and a lumen just big enough for a screw to go through. The dimensions of the hose are: OD = 13 mm, ID = 6.5 mm and Wall = 3.25 mm.
So I cut two rings from the hose, about 3 mm wide, as in the pictures below:
Then I pushed the ring in the hole, top side first:
and then I pushed the bottom edge to fit in the hole. It needs some force to get it in but it's not too hard because the hose will flex:
Then go back to the top side and push it in. The ring is now completely in the hole:
Keep working the ring with a flat-head screw driver from all sides until it slides to the bottom of the hole:
Careful not to go too far and come through the other end! I left about 1 mm to the end of the hole and this is what it looks like on the other side of the handle:
At that stage I put a screw though and pulled with all my strength but the ring would not move at all. It was in there well and truly stuck. As an added measure I added a blob of Araldite on this end (making sure not to obstruct the lumen of the hose where the screw will have to go through) and that was it:
I hope that I will be as pleased with it when I put the handle back on the car.