So I’ll be in Taiwan for a few months soon due to study commitments, but have no access to skype to chat with family back home. Anyone know of any wangka’s/internet cafes open late they can recommend around these two cities?
There are hundreds of Wangka’s in Taiwan but I can not tell you exactly where they are. 
You can always use any Macdonalds/Starbucks 
Nanyang Street, Taipei. Near the main railway station. There are several in the area. Mostly used for gaming. They used to be awful, smoky places, but I think the city’s ban on smoking in indoor public spaces has improved the situation tremendously.
There are hundreds of Wangka’s in Taiwan but I can not tell you exactly where they are. You can always use any Macdonalds/Starbucks [/quote]
There are?! There used to be, sure, but I thought they’d mostly disappeared now. Yes, there’s wifi everywhere, but I think the OP is also looking for a computer to use.
My favourite one in Taipei is opposite the old Sogo at Zhong Xiao Fuxing.
It’s a complete glass fronted place, but the shop itself is down stairs (so if you look in the window, you can just see a staircase going down) and usually has a cardboard cut out of an anime girl by the door.
I feel like this is a troll post…
However, monokuro’s suggestion is good, the place is expensive, but I hear it’s pretty good and clean.
There’s one behind the Zhong Xiao Sogo that is almost directly across the street from the one monokuro mentioned. It may be a little cheaper. The ground level is a Ttesports store and the downstairs is the internet cafe.
It’d be helpful if you could provide your location in Taipei or Kaohsiung. If you’re studying, I assume NTU or NTNU? If so, why not use campus computers? Unless there aren’t any available for students to use?