I've read the answer you refer to and tried the method on this particular image.
The problem in this case is that the white contour is broken by the two eyelashes in the right side.
If you click the white contour with Magic Wand Tool, you won't select the part between the two eyelashes. You can Shift + left click the missing part to add it to the selection, but the selection will still have a hole in it.
So when you enter Quick Mask Mode and try to click the inner area, it doesn't work because it's connected to the outer area.
A solution could be to use the Brush Tool to paint in the part missing from the selection with white.
Now the inner area is separated from the outer area and you can continue with the method described in the other answer.
You might want to add the two eyelashes to the selection afterwards, so you don't just cut them off. This could also be done in Quick Mask Mode with a white brush.
You probably need to put a little more effort into it than I did here.