I just stayed in Taipei for a month and visited Kaoshiung for four days. It seemed to me that many restaurants close very early in Kaohsiung and it was harder to get around because there are only two metro lines. They have a bikesharing system like Taipei, but it doesn’t work with the payment card for Taipei, so I didn’t use it. So overall it didn’t feel as “easy” as Taipei.
Regarding the internet speed; I guess it depends on what you define as “fast”. The wifi in the hotel I was staying in was worse than 4G on my phone. But I had the same in Taipei in a few places as well. And 4G was good enough for video calls, which is the most demanding use I had for my internet connection. If you need high bandwidth to transfer large files, this might not work for you.
Personally, I would probably get bored staying a whole month in Kaohsiung. If you haven’t been to either, I’d say go to Taipei and do a weekend trip to Kaohsiung. It’s super fast to get there by bullet train.