In the OP's case the structure isn't an extension, so is not covered under exempted development.
Getting the structure removed is the only recourse here. Lodging a complaint to the LA is free.
Being the sceptic that I am, I looked into it myself and true enough, building extensions up to 40sqm did not require PP....unless....the extension building has a window that overlooks another property whose boundary is within 11metres of the extension.
You may have some scope in that regard for some monetary compensation, but I would expect between legal costs and challenges (and peace of mind) it would not be worth it.