If you're looking for budget accommodations in Taipei, check out Eastern Star Hotel. It's near the MRT station and shopping district.
Here's our family's experience at Eastern Star Hotel...
EASTERN STAR HOTEL
11F, No.209, Sec. 4, Jhongsiao E. Rd., Da-an District, Taipei City 106
Telephone: +886 2 8773-6638
Check out low rates and availability from Agoda: Eastern Star Hotel (aff link)
From Taiwan's L'Hotel de Chine Group comes PALAIS DE CHINE, a 5-star hotel inspired by the sophistication of Eastern Palaces with the elegance of their Western counterparts. It's a visual treat to stay here. Best of all, it's located in the heart of the transport hub in Taipei. :)
Here's a photo essay of our stay in Palais De Chine...
PALAIS de CHINE Hotel Taipei
No. 3 Sec. 1 Chengde Road Taipei 103 Taiwan, R.O.C.
Telephone: +886 2 2181-9999
Check out low rates and availability from Agoda: Palais de Chine Hotel (aff link)
You can easily get a 30-day multiple entry Taiwan visa for FREE if you have a valid visa for the US, CANADA, UK, JAPAN, or SCHENGEN countries.
Otherwise, you have to apply for a FEE via the Taiwan Economic and Cultural Office located at the 41F, Tower 1, RCBC Plaza in Makati City.
For a HASSLE-FREE visa application, I highly recommend Jeron Travel.