Travelodge!
Now I know these are a bit "soulless" but they are cheap (£29 for two of you), allow you to take your bikes into the room and are clean and welcoming.
There are also a lot of decent pubs and cafes in Durham to eat in.
The one in Durham is only a few minutes walk from the town.
In fact here is my bike ensconced in said Travelodge!