To be honest, I think your plan sounds fine. I live in Sheffield, and train fares to Leeds, York, Nottingham and towns in a similar radius are pretty cheap. And I came over as a student twice and loved being able to take daytrips while at uni and not having to lug my stuff around all the time. And if the goal is more to experience life in England (inasmuch as you can do that in a week's time) rather than sightseeing, then staying in one or two places sounds sensible.