Seems like they broke it.
Ctrl+Shift+K doesn't work when the focus is inside the compose window. It works if you click into Gmail outside of the compose window.
This might be a workaround for you:
Press Ctrl+Shift+K to change your language setting
Then open the compose window
Of course, that's useless if you want to switch while you are typing ...
The link on the question now redirects to Google Input Tools. The list of supported languages is on https://www.google.com/intl/en/inputtools/help/languages.html. For Japanese it indicates that offers Input Method Entry (IME) and Handwriting.
From Android Enthusiasts : Inputting katakana on Google Japanese Input