First of all, the period you mention covers the start of the Tour de France, which (strange as it may sound!) starts in Leeds on July 5 and the route goes west and north that day, the second stage starts in Harrogate and goes to Sheffield and then heads south on 7th to London. This means that accommodation for a LONG distance in all directions from Leeds and especially north and west of it is VERY hard to come by at this point. I know because my family and I will be there and I was trying to help a friend find something. So, I think getting close to Hadrians Wall, ie: the Lakes, is a good idea, rather than trying to see things on the way.
Real answer though, what do you want to do? Are you an active, fit, outdoorsy type of person? If so, then you can use public transport to get around and do some of the most incredible walks you will ever do around the Lakes. Truly spectacular. I second the suggestion of Keswick, fantastic little town, some nice walks not far away (Walla Crag, Catbells, many others).
If you are more into cities then this isn't the best plan, Durham suggestion would be better.
Cheapest accommodation would be youth hostels, check the YHA website, but you don't say what you really mean by "cheap" accommodation.