This web site

http://www.chinese-names.net/doc/chinese-character-fonts

illustrates, by name, the various traditional forms of the Chinese script, including Kai Ti.

With that information, from resources like

http://www.wazu.jp/gallery/Fonts_ChineseTraditional.html

you may be able to select another suitable font, although the only one there with "Kai Ti" in its name is one with accompanying "ruby" for every character - not useful for general purposes.

However, the AR PL ZenKai Uni fonts also appear suitable.

when we select it the font doesn't display in Chinese

Can you elaborate? What happens when you apply the font to a Chinese text? In what form you receive the Chinese texts to import in InDesign?

On OS X 10.9, only Kaiti SC Black has fsType set to "restricted embedding"; the other Kaiti fonts are set to "editable embedding"

Font fsType

Kaiti TC Bold 0x0008
Kaiti TC Regular 0x0008
Kaiti SC Black 0x0002
Kaiti SC Bold 0x0008
Kaiti SC Regular 0x0008
STKaiti 0x0008

For the meaning of 0x0008 and 0x0002, cf http://www.microsoft.com/typography/otspec/os2.htm#fst The embedding problem with the regular and bold was with older versions of the OS.

If, when you copy text from http://zh.wikipedia.org/wiki/中國 and paste it in InDesign, Kaiti SC renders it properly, then the problem is not due to Kaiti but most probably to the encoding of the file you were given. For instance, I took the text http://typophile.com/files/text-gb2312.txt from www.bmo.com/s_chinese which is GB2312 encoded and saved it as plain text. If you open the .txt file with your browser, you will most probably need to choose view > text encoding > Simplified Chinese (GB 2312) so that it displays properly.

Added: the very same line of Chinese text can also be found in text-bm-utf8.txt; this time however you need to select the encoding Unicode (UTF-8) to view it. To get that new file, I just opened with TextEdit the previous GB 2312 encoded text and saved it in the UTF-8 encoding.