It does sound like you should have been employed but that is really their problem as they should have been deducting PAYE and paying employers NI. You need to submit a self assessment tax return with your business expenses on it and see how things turn out. The worst that can happen is that you won't be allowed any expenses and will have to pay the tax on the full amount you earned.
Have a look at the HMRC webiste but you should be able to claim for the business proportion of your petrol, car lease, servicing, tax, tyres etc. You'll have to include the amount you were paid as mileage reimbursement as income though. You'll also be able to claim a proportion of the running costs of your home and the business element of your mobile. Don't try to claim for anything you can't justify or haven't got receipts for though.
You'll have to put your income pre August on the return as well as your JSA but you'll get the full years allowance to offset against your net income (ie. earned and net profit) and you'll ahve already paid teh tax on the employed bit.