For commuting over that distance in what could be nasty weather I would go for a hybrid with full mudguards and some continental gator skin tyres, possibly 28mm.
---------- Post added 19-08-2014 at 16:28 ----------
For a commute like this I would go for some hybrid with full mudguards and continental gator skin 28mm tyres.
Going through Castleton to get to Chapel means Winnats or Mam Tor, Both steep. You could get to Chapel via Peak Forest but it's a longer way round.
Check out the Tissington trails around there, you might find a better route for some of the journey.