There used to be a Starbucks on the 35F of Taipei 101, open to the public, but required a reservation. There was also a strict minimum spend of 250 NTD and a 90 minute time limit. I had a chance to test it before they closed the Starbucks during the end of 2020 (maybe it will open again in the future?)
You need to call and make a reservation, at least 1 day before. Arrive at Taipei 101 around 10 minutes before you reservation and then a staff member will escort you up to the 35F. The Starbucks itself is very small. There are mostly counter seats along the windows with good views, as well as some smaller tables. There was only 1 power outlet but it wasn't near any of the windows. The WiFi was unfortunately quite slow as well.
A fun experience, but perhaps not the most workable place in Taipei.