If you already have 2 days booked, I would do a day trip somewhere, maybe Hualien, and then spend the rest of the time in Taipei. Driving around will just eat up so much of your time, so I recommend staying in one place for longer. Taipei is the cultural hub of the country and has so much to see. Visit some of the night markets. Night markets are an amazing experience, and a must-see during a visit to Taiwan IMO. Your kids would probably enjoy the Ximending night market in the eastern part of Taipei. Its a massive shopping area, and is the largest pedestrian area in Taiwan. Its also arguably one of the culture hubs of the city. Since its more tourist friendly, there will be more food options for less-adventurous palates. If you want to experience a more authentic night market, my favorite is the Tonghua night market in the Da'an District. However, you can't go wrong with many of the others. There's so many with different atmospheres and food options. You could easily spend all 6 nights visiting a different night market. Some other recommendations:
- Taipei 101.
- Minatures Museum of Taiwan.
- Chiang Kai-shek Memorial Hall.
- The Bao'an and Confucius Temples.
- Hike Elephant Mountain.
If your kids are into sports, going to a baseball game would be cool. They have a professional league, CPBL, and the season will have started by the time you're there. Taiwan is such a great country. Have fun!