Brave the crowds through the night market in ShihLin (it's on the red MRT line, JianTan station).
Head up YangMing Mountain and enjoy a coffee or tea with views overlooking Taipei.
Do as the locals and book into a KTV (karaoke) place (such as the popular CashBox chain) with friends and sing your heart away!
Wander the malls (seasonal).
Rent a Giant bike (multiple locations around the city, mainly along the river) and take a nice easy ride.
Eat, eat, eat! Taipei has a lot of really good local foods, and excels in high-end Japanese cuisine at relative prices that you won't find anywhere else in the world.