I think you are on a hiding to nothing with this one. Barbel have reached as high as Chatsworth to my knowledge but flyfishing in most parts is not allowed or you need a particular ticket.
Lets start above Derby; The Railway Club has good fly water with some fast runs at Duffield. I think the season ticket is about £35 but not sure. Above Milford there is some fast water on the Belper AC stretch at Bridgefoot and day tickets are available but never seen a barbel caught on fly as it's mainly a coarse club.
Above Ambergate there is some good fly water but LB Plastics water and no tickets available unless you work there! Ripley AC have water above but only really one stretch suitable for barbel below Whatstandwell Bridge. Above The bridge short stretch from Derwent Hotel could well hold fish.
Above that is a short stretch controlled by Prince Albert (but not for long if what I hear is right) so season ticket only.
Cromford Flyfishers above that again on season tickets but if you know someone who's a member it could be worth a try.
Derby County Angling above Cromford bridge should hold fish but I have no details of any coming out. Perhaps someone knows more about it than me.
Matlock Angling above but no fly fishing in the parks. Forget the canoe stretch


in Matlock bath.
Other clubs above matlock where barbel have been caught but not to y knowledge on fly - again I may be wrong.
Rowsley fly fishers - don't know. Fish should be there.
Chatsworth - we took one of 8lb post spawning last year - just lifted it out with a landing net weighed and returned so fish there but I have never hooked one on that stretch.
Further up towards Calver I don't know they may have ventured that far.
Try the usual bugs and nymphs with some weight or gold heads.
Hope this helps
Tim