Hi, as a mum in kaohsiung, hope what I experience can bring you help.
Most of families in Taiwan are double-income family, esp. in southern Taiwan, work from 8-6, therefore, schools/kindergartens that provides 8 hours day-care in order to meet their needs, some are even requested by the parents. it's kind of social custome and we all think this is quiet normal. Of course, if you wish to have your kid taking half-day lesson which is morning session only, you may ask the kindergarten to see if they have such an exception. But be aware that the tuition might not be a huge different from the whole-day.
I know the school you mention, that's an international school which mostly enhance English more than Chinese and it locates outside of the town. If you are coming here to teach at that university,that school has kindergarten and all the way to university, it's a good idea to enroll your kid at their school, however, if you will work in Kaohsiung downtown, I suggest looking for other schools in the town. The distence from that school to downtown is quiet far. There are various schools in town which can also meet your needs.
hope above description can bring you some ideas about the kindergartens in Kaohsiung. If I can be any of help, pls don't hesitate to ask.