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@The_Kindness_of_Trees
You will need something like this:
http://www.ebay.co.uk/itm/CYCLE-REAR-CASSETTE-FREEWHEEL-SHIMANO-7-SPEED-14-to-28-GEAR-TZ-541-/181237472594?pt=UK_sportsleisure_cycling_bikeparts_SR&hash=item2a32985d52
Your rear spacing will be 126mm (probably) so 7-speed is high as you can go (I think), which should be ample anyway. Maybe check with the seller to make sure it's compatible.
The only other thing to then consider is whether or not the chain will need changing to accommodate the increased gear range (anybody?). Either way, it should be a fairly straightforward job.