Since you got no payslips you really do have a problem. I suspect you probably didn't get a year end P60 either which summarises your total pay and tax for each tax year.
I think you will just have to wait for the outcome of the tax office investigation and let your lawyer deal with it. If you had payslips or P60s you would be in a far better position. You may have to try suing your former employer for the fact that you can't get any jobseekers allowance. The fact that you have a contract of employment is a good thing. It shows that you were an employee and not working on a self-employed basis so I can't see how the Revenue could seek to tax you.
In order to try to force the Revenue to give you credit for national insurance contributions which will help you claim jobseekers allowance, you might want to contact www.taxaid.org.uk which is a charity that specialises in helping people with tax problems and little money to pay for professional help. Your sort of case might be outside of their remit but it's worth a try in my opinion.
As far as the jobseekers allowance itself is concerned, there must be appeal procedures you could follow and I would suggest you get in touch with your local Citizens Advice Bureau for advice in this regard.