Did you appeal the planning refusal?
Appealing the enforcement notice is not too dissimilar to basically putting your LDC information into the enforcement notice and letting the inspector assess it - in effect he will be assessing whether or not the development requires planning permission (depending on what ground(s) you appeal on)
You may well want to get professional advice to give you the best chance (Info from the Enforcement investigation, Other appeals, a proper breakdown of the legislation, etc etc)
Best of luck