I live near Ipswich and I know that most of the cheaper B&Bs charge around �30-�35 per night. There's very little available at lower prices. The only place which I know of that offers really cheap accommodation is Jack's Cafe. It's located about 10 minutes walk from the railway station and is really easy to find by road. (It's on a corner of the main A1214 approach from the south.). To walk into the town centre takes about 15 minutes. I'm not sure of their current prices but I know that they've always been the cheapest place in town. The address is 4 Hadleigh Road. Phone 01473 253 388.
Another place to try might be Duro's Cafe. I know that they do B&B but I don't know their prices. The location is about 15 minutes walk (at a fairly leisurely pace) from both the rail station and the town centre. By road, it's just off the main A137 road into town. The address is 67 Wherstead Road and the phone number is 01473 601764. I've got their business card in front of me, which also gives two mobile phone numbers but the card is fairly old and these may have changed. The numbers given are 07834 172468 and 07834 172500.
You've not said when you'll be visiting Ipswich but I advise booking as far in advance as possible. I once tried to help someone find a room for the same night in Ipswich. I phoned 52 hotels and B&Bs but, despite her being prepared to spend up to �100 per night, there was nothing available. (She had to stay in Felixstowe instead). Ipswich is a major business centre, with several big insurance companies having their headquarters in the town. Additionally, the BT research centre (which is almost a town in itself) is nearby. These places have loads of people visiting them from elsewhere in the country and the B&Bs fill up very quickly. (Many of them charge �60 to �70 per night because they know that there will always be people wanting accommodation). The cheapest rooms available at short notice often cost around �