+1 on the suggestion for Din Tai Fung. You definitely have to try the steamed buns (or 小籠包, pronounced xiao-long-bao).

I would also try to find a place with good beef noodle soup (if there are any suggestions from locals, I'm all ears!). I think people have recommended 牛爸爸 (or Niu Ba Ba) but I haven't visited it before. I think it's worth a try.

I'm not a big fan of night markets. I prefer sit down restaurants serving traditional Chinese/Taiwanese cuisine than street food.

I would consider Taipei to be more of a food destination than anything else. You can do all the touristy stuff such as the 101, CKS/SYS Memorial Hall, National Palace Museum, Yangmingshan, etc. but I think your impression of Taipei will be much improved with all the eating you'll be doing. Just my two cents - coming from a Taiwanese American who was born in Taipei.