Sounds like so far you are planning to only stay around the Taipei area. It's a good starting point, but there's a lot more to see outside of Taipei. It depends on your budget, time and what you prefer to do, outdoors, museums, nature sightseeing, historic sightseeing, or night life.
The only problematic item on that list I guess would be going to a spa (hot springs). The weather is going to be humid and hot in June, Taipei city would be particularly hot. Spa might be more enjoyable in the winter. In the summer spa is enjoyable if you hiked a lot and developed sore muscles, then it is a good way to relax.
Other than that, your list looks fine. Danshui definitely is worth going to. There's a mangrove path near Hongshulin station (紅樹林站), which I think is worth going if you don't mind the summer heat. Remember to visit Fort Antonio and fisherman's wharf in Danshui.
In any case You might consider some excursions to higher sea level to escape the heat. Taiwan has some great alpine landscapes to offer. The downside is that you would need to find some form of transportation.
Some ideas for you to consider, if you can only rely on public transportation, you can checkout this free government run tour bus service.http://www.taiwantrip.com.tw/Besttour/L ... lang=ko-kr
I think for a day trip up to the mountains, you could give Xiaowulai (호행노선) a try.
The east coast is amazingly beautiful, but You probably would need to spend all 3 days there to worth the trouble getting there. 꽃보다 할배 perhaps didn't do Taroko justice, it certainly deserve more screen time. If you enjoyed your trip around Taipei, you should definitely spend another 4 day vacation in the east coast.