Help with taking a company I freelanced for to court for not paying me in full

I worked for a company in 2009 and 2010 and I provided all the necessary documentation for proof of work I completed. This documentation was then used by the company to invoice their client a local authority for their money which they received. Due to ill health, I had money owed to me by the company from 2010. I invoiced for this money in November 2011 and the company is saying they can not pay me in full due to the length of time that has passed. They also say they do not have the paper work. A few weeks back I sent an email requesting payment in full otherwise I would go to the small claims court. They replied saying they would pay me for part of the work as a final settlement. I rejected this final settlement as I would be out of pocket.

I would like to claim this money back through the small claims court it is under £5,000.

I have emails to proof I have been in communication about this issue trying to resolve it with them and the programme manager who worked at the company is happy to confirm that I submitted all the necessary information at the time.