You can pretty much write off all of the east and west midlands!

One of the only good things about Grantham is that it has a main line railway station, meaning that a direct train to London takes only just over an hour and they're quite frequent.

I don't know how much time you've got or what your transportation arrangements are like but York is well worth a visit. Yet again, the train looks doable with only one change at Wakefield (you can probably safely avoid that as well!) and a travel time of just under 2 hours.

Manchester and Leeds aren't too bad, but I'd say to see as much as you can without expending too much effort, a car would come in handy.