I went to the Kenting night market earlier this year. The only thing remarkable it about is the alcohol available, lots of beer and cocktails. Almost any major city night market has a greater variety of foods and snacks. But if you are there at night, you should go. There's not really any choice.

Kenting is tough for me to recommend. For Taiwan, it is one of the few, and therefore among the best, places for a sand beach with water activities. But frankly there is just no comparison with the beaches in SE Asia. Caesar's Park is a solid choice, especially with kids. Eluanbi Park, south of Kenting, will be good for short hikes, be sure to explore the trails leading from the grassy area to the shore. Very cool rock formations there.

OP, you are really packing in a lot for your trip! I think you are a bit light on time for Kaohsiung, it's an interesting city for visitors. While you have the island of Taiwan really well covered, but there are like 80 other islands. Maybe with kids it would be good to take the ferry from Kaohsiung to Xiaoliuqiu for a day visit. There's a lot to do in and around Kaohsiung.

Taichung is getting an Intercontinental but it will not be open yet when you visit. Le Meridien Taichung is a nice hotel with good food in an unfortunate location. Kaohsiung and Tainan have some areas that are chic shabby, but this area is just shabby. Totally safe, but in need of serious urban redevelopment. I have spent a lot of time in Taichung and frankly it is not a city I can find a lot to recommend for overseas visitors.

But since you have a car, if you want to do consider some things most visitors don't get to see, take a look at the 921 Museum 921 Earthquake Museum of Taiwan-National Museum of Natural Science (nmns.edu.tw) which has some interactive displays as you walk through the ruins of the school. Or Shigang Dam Shigang Dam 石岡水壩 | Spectral Codex. You can rent bikes or just walk across the dam to the other side of the river. The dam is the most visual, demonstrative remnant of the 921 earthquake, The change in elevation due to the underlying fault was around 15 meters breaking the north end of the dam into pieces and putting them at a 35 degree slope. They remain largely unchanged today. Otherwise the area is now a park.

On the lighter side, a bit further up into the mountains outside Taichung the Xitou Monster Village is always popular with kids and they have some really nice resort type hotels. This village was a Japanese mining and logging area, and they took the Japanese theme in a totally unexpected direction when they redeveloped it for tourism. It is directly adjacent to the Xitou Scenic Area but after Taroko the most impressive aspect will be the size and height of the trees. Which you can also see in the Monster Village. xitou monster village - Google Search

Just some suggestions of some unique things that are accessible for you with a car. Please drive safely and ignore other drivers not following traffic regulations.