Hi Dave
The Google Analytics In Page view actually does not show how many people click on a specific button but rather how many people visit the page the button links to. As you know Analytics only, at least by default, register pageviews, not clicks on links or buttons.
So if you would have more than one link on a page pointing to a specific page you would not be able to see what button actually got the clicks. The number on both of them would be the views on the page they lead to.
In your case the button is implemented with a somewhat "odd" instead of a "normal" button. I would suspect that the in page tools is not configured to show statistics for this tag. BUT, that does not mean that the data is not there. Its just not shown in the in page view.
You might want to consider swapping it to a href or submit button and then add the image with some simple CSS. This would actually be "better" HTML as well. Maybe something like:
input.delete { background-image: url('bin/images/common/delete.png'); width: 20px; height: 20px; border: 0; cursor: pointer;}
Hope this helps
Fredrik