Loose Leaf Tea in Taipei
There are lots of independent tea shops all over Taipei that sell loose leaf teas. Ten Ren is probably the most well known chain that also sells drinks, and has wider global appeal.
But in actuality there are other places that probably offer better quality teas. Look into Wang De Chuan, although they only sell tea from China and Taiwan, great quality stuff.
They may still have a small outpost in the basement of Sogo, and you might be able to find an outpost for Lupicia as well (which also has international teas, but their Chinese tea selection is not going to be as deep as WDC).
There is also a much smaller WDC outpost right around the bend from Ten Ren Tea station in Hysan Place Causeway Bay (where Eslite bookstore is located)