12 days/11 nights Taiwan/Singapore + maybe HK?

Spending 12 days/11 nights in October between Taiwan/Singapore. We are flying into Taipei direct via JFK. Will fly back to JFK from Changi. Plan to spend 7 nights in Taiwan and 4 in Singapore. We are wondering if we should stop by HK as well? It would be broken down to 5 nights Taiwan, 3 nights HK, 3 nights Singapore. This is our first time to all 3 areas.

What do you guys think?