Shilin Night Market is always highlighted by travel blogs and websites. I stopped on the way back from Beitou to see it myself. To me, it was just another night market especially when I had nothing to buy.
I was there quite early so there were not many people yet. Therefore I could move around easily. If you are hunting for local delicacies, I guess this should be the place.
You could also try your luck at the game stalls.
In the end, I just bought some preserved fruits to bring back to Malaysia - mango, guava and the other one I could not remember.