Actually by February things can oftentimes be looking up, the last few Februaries have promised (only to decieve) good weather through spring.
It has been quite mild over the last few years, which has fooled the wildlife into early than usual awakening, with daffodils and other spring bulbs coming up too soon and some winter dormant animals to rouse themselves, and flowering trees to produce blossmoms.
This has been taken by the doomsayers as a sign of global warming.
It could easily turn into the more normal February, the ground is pretty well sodden and heavily muddy so I would not advise much in the way of walking outings, and if the weather is bad you may wonder what happened to the term ‘daylight hours’.
Few places you might try if they are open, industrial heritage is big in the north of England, rate this as PG
http://www.beamish.org.uk/digest.htm
I’m sure there will be something here for you,you decide,
http://www.manchesteronline.co.uk/tourist/museums/pages/ancoats.html
Over the penines is the National railway museum, rate it as PG
http://www.nrm.org.uk/html/home_pb/menu.asp
If you can catch a live demonstration of combat then this is also PG
http://www.armouries.org.uk/leeds/index.html
If you go to Whitby then just up the road from there is Robin Hood’s Bay, and since you state you like places off the beaten track this would be ideal, I think this place is better in bleak weather than in summer with all the tourists, but I don’t expect you will find much open, which makes it all the better IMHO
http://www.robin-hoods-bay.co.uk/
I think York is a must see, make sure you get a good guide.
http://www.york-tourism.co.uk/visitors/visitors_frameset.cfm
If you are not going over to Yorkshire then this may well be worth a call
http://www.chestercc.gov.uk/tourism/