The National Palace Museum is definitely on my list!I was in Taipei many many years ago, so no idea where we stayed, but you have got to visit the National Palace Museum - I was 16 when I went there, and it is still one of the most amazing museums I've ever seen.
And eat all the food! So much yummy food there - it was the first place I ever ate bao and I'm still obsessed
I'd love to go back and explore more of Taiwan!