Travel advice for Taipei, Taiwan

I'll be visiting Taipei for a little over a week in mid-January. I'm going to attend a conference, but have most evenings free and will be staying a few days after it ends too. Looking for good recommendations on activities in the city (museums, sights, etc.). I've heard Taipei has a very good nightlife, but unfortunately I don't know if I'll be able to experience much of that due to with whom I'm traveling.