One place I'd always recommend people to visit, even if it is just a quick one, is Ximending.
If you don't mind not going to a night market, I'd pick this over Shih Lin Night Market as Shin Lin seemed a little too touristy for my liking.
Ximending stores opens till pretty late as well, it was pretty crowded even at 10pm+
Ximending is like a place where there is a little bit of everything. You have sit-in like cafeteria/restaurants that service local delights like dumplings, egg-and-oyster omelette,
rice vermicilli,
shaved ice dessert (made from frozen milk actually, not water) and more!
Also, you have hawkers selling food off little push-carts! Hope that you do not have a weak stomach, as some of the most delicious food I've had in Taiwan came them these stalls! Once in a while, the police would chase them off and they would play a little hide-and-seek game, only to eventually return to the area
On some nights, you have
artistes like this one (I just saw him a few days ago when I was there, you have to watch this video) who is simply amazing!
Of course, there are also stores selling all sorts of stuff like clothes, bags, shoes, accessories etc.
I love Taipei
