Rye House is probably the closest track to you although Whilton Mill in Daventry and Sandown Park are probably quite close aswell.
Rye House is open for owner driver testing at weekends and I think its about £40, they also do ARKS tests and driver training although I'm not sure about the cost.
Also they hire out a full blown Rotax Max as a taster, I think the cost is £130 for an hour and the Kart shop also sell second hand karts.
Have you considered Club 100 and Easykart? I've attached some links below.
http://www.hoddesdonkartclub.co.uk/
http://www.club100.co.uk/
http://www.easykart.co.uk/
http://www.rye-house.co.uk/