For me, getting outside on weekends is key. And going from city into the parks/mountains/coast/green spaces can bring the temperature down several degrees. For many years, a couple friends and I just pick a mountain park or out of the way scenic area on the coast and head there for a day of hiking/walking followed by a nice meal. Keelung Mountain near Jiufen is one example http://www.slightlyastray.com/keelun...ike-in-jiufen/

Note that this is not Alishan style hiking. These hikes are doable in several hours of strenuous exertion but many of the times We see very few other people. Certainly it is not like hiking Yangmishan where I have run into countless people I know climbing up and down the mountain.

Bitou is one of my favorites https://bikehiketaipei.wordpress.com...-cape/?wref=tp On the coast with nice views, some nice resting places, and near a couple of great seafood restaurants where everything on the menu can be selected from the tanks outside.

Want something more challenging, try Teapot Mountain. https://bikehiketaipei.wordpress.com/teapot-mt/

As said above, last week we rented motorized bikes and rode along Fulong Beach after going through the tunnel under the mountain. https://bikehiketaipei.wordpress.com/fulong-loop/

The same day we visited Hutong cat village and supposedly there are some nice hiking trails in the old mines but it was raining so we didn't stop.

Not sure exactly what is on this site (and it is a bot old) but it is called Taiwan off the beaten path so should give you some ideas. http://www.taiwanoffthebeatentrack.c...ober-22nd23rd/

Note that mostly I do day trips from Taipei leaving early in the morning and coming home well into the evening usually full after a banquet accompanied by more than a few shots of Kaoliang or whiskey, but I have access to a car and don't always depend on public transportation. And with your daughter, the second part is probably not recommended.

But if you can get away, you can go diving in Kenting but I have never done that and have friends that have dived around the world and say that it was okay but certainly not world class.

And for a nice two or three hour diversion, walking around Lotus Pond http://www.myseveralworlds.com/2013/...uthern-taiwan/ or Cijin Island in Kaohsiung is nice. https://trip101.com/article/day-trip...s-in-kaohsiung