Check out the following budget hostels in Taipei

• Taipei House International Youth Hostel - Songjiang Road
• Holo Family House - opposite Taipei Station
• Fortuna Hostel - convenient location near the Songshan Railway Station
• World Scholar House - near the Zhongxiao-Xinsheng MRT Station
• Chocolate Box Backpackers - Shi-Da/Guting area
• Eight Elephants Hostel - Shi-Da/Guting area
• Taipei BackPackers -Ximen
• Mandarin Hostel - two locations in Taipei
• Happy Family Hostel - right near Taipei Main Station
• Camel's Oasis - near Taiwan Democracy Hall (CKS Memorial Hall MRT Station)
• Taipei Hostel - near the Shandao Temple MRT Station
• Amigo Hostel - In Yonghe near the Dingxi MRT Station
• International House of Taipei - located in Xindian
• Hillside Hostel - in the east of Taipei

Hostelling International (YHA) has several hostels and an office in Taipei. The main hostel is located in the K-Mall at 50 Zhongxiao West Road, opposite Taipei Main Station. More details here.

There are also many hotels in Taipei and some of them have prices that are very affordable (less than NT$1,000). Often the signs are only in Chinese though and very often they are "love hotels" advertising rates for both short stays and overnight stays. There is no reason you can't stay in one of these hotels though. They may even be willing to give a discount for long term stays.

You can find a good list of Taipei hostels at Travellers Point. You can also make online bookings there.