Unfortunately, looking at that, they're probably within their rights. You hoped to get some extra money and they don't know that you didn't.
When you say people in the company knew, did your line manager know? If so, I would go back to them and say - look, I didn't hide this. My manager knew and never said it was a problem. If your colleagues or other managers knew, it will be more difficult to make your point.
Also, they'd argue you did use work resources as you wrote about it on your work email!
Do bear in mind that you can go to CAB but you should think about this before entering into conflict with your employer in any big way - these people may be asked by future employers for a reference: is it worth having a cloud hang over them, for a months pay which you may not legally be entitled to, anyway.