I spent a week getting a stomach ulcer treated at the Zhongshan hospital, in the Lanes behind the southeast corner of the DunHwa-RenAi circle.
I had only been in Taiwan for a couple months at that time, and it was the only hospital I knew (from living next door to it for my first 2 weeks in Taiwan), which is why I ended up there (through the ER at 4AM in the morning, no less). It's a smaller hospital, and I'm told that the word on the street in Taipei is that it's pretty good, even though it's not one of the large well known hospitals. Several friends have told me that it used to be a private hospital for government officials and the wealthy, so it must be pretty good.
Anyway, I thought the care I received was pretty good, and the followups were very professional. Then again, that was the first time in my life I've ever been to a hospital in any country, so I have nothing to compare against.