Taipei is a great place to study Mandarin. There are a number of choices available to you, from small private institutions to large public ones. If you will be staying in Taiwan beyond the 2 month visitor visa, be sure your language institute can provide you with a student visa. Not all language schools can do that anymore, as the government attempts to crack down on illegal teachers.

Even though Taiwan uses traditional, non-simplified, characters, many schools teach Mainland Chinese characters in their programs. Be sure to ask for what you want to learn. There are courses offered at NTNU (National Taiwan Normal University), usually refered to as ShiDa, through their Mandarin Learning Center. Other well-known program is at TLI (Taipei Language Institute).

There are numerous other programs available. For a list you can read this Transitions Abroad article.