Here you can see a close-up of the entire lever assembly attached to the back door of the housing.

Here you can see in detail how the covering piece has fractured around the two screws, it can no longer hold the lever such that it can apply the necessary pressure to the button, instead it now flexes where the fractures are.

Here are all the parts disassembled and you can see better the fracture patterns on the cover part.

After I had disassembled the parts, I concluded that it would be really easy to make a replacement part for the broken cover myself by cutting a similar piece out of a polycarbonate sheet, and I happened to have all necessary tools available. Since it seemed likely that the original part made of 1.6 mm (.062”) thick sheet would soon break in the same way again, I decided to try thicker material, 3.2 mm (.125”) polycarbonate sheet.
Here you can see the part that I made myself, it only took a few minutes make.

On next image you can see the new part attached. I decided to add one more screw to the left bottom corner, as it seemed that the lever caused the greatest pressure at this point.

Unfortunately, now the cover became too strong for the screws to hold it. When testing the assembly, the lever caused such a big pressure that two attachment screws broke off the housing completely, chipping off pieces in the area where they were attached to. The resulting damage can be seen here:

I will be sending the housing for repair to Ikelite soon. Maybe after the next weekend's diving trip, which might the last one of this season.
As a temporary workaround I am using a long thick weaving pin to push that button. Not so convenient, but better than nothing 
Higher resolution pics here:
picasaweb link