I just had a stay in the Regent Taipei in Oct 2019. Personally, I find the breakfast to be good. It can get busy but typically the queue clears quite fast.
Food wise, there is great selection of local Taiwanese / Chinese dishes, Western, Korean and Japanese food. I have not tried the food at the lounge and as such, I can't comment which is better. There are also juices, fruits and desserts to choose from.
Enjoy your stay.