Below is a map with hotels and B&Bs in the area, as well as the Cathedral and Athenaeum.  We've tried to be as complete as possible, but there may be some missing: let us know if you know of a good one that's missing.  From the long list below, we have experience of (and can recommend) the Chantry Hotel and Best Western; if you have the budget for it, The Angel is certainly worth it, especially considering how close it is.

Walkable from Cathedral / Athenaeum

The Chantry Hotel (£78-£132)
The Angel Hotel (£85-£375)
The Fox Inn (£119-£139 few rooms available)
The Abbey Hotel (£83-£90)
Regency House Hotel (£90+?)

B&Bs

The Dog and Partridge Pub (9 rooms, £70-£80)
Northbury House (3 rooms, £65-£90)
The Bushel Pub (£89.95)
Number 6 Orchard Street (3 rooms, £45-£55)
The Old Cannon Brewery (Pub) (£90-£120)
Ounce House (£90-£130)
Sycamore House (2 rooms, £60-£80)
83 Whiting Street (4 rooms, £45-£70)
The Black Boy Pub (5 rooms, £40-£80)

Long walk / short drive

BEST WESTERN Priory Hotel (£60-£76)
Quality Hotel (£85)

Further afield

The Suffolk Golf and Spa Hotel (£62-£89)

Cheaper but further

Travelodge Ipswich (Stowmarket) (from £31) 18 mins
Travelodge Barton Mills (from £31) 23 mins