I installed both Korean and Japanese using the Language Support window, and expected Korean (Hangul) and Japanese (Anthy) to show up in the input sources for Text Entry Settings (having both packages installed). However, neither shows up.
The comments in Writing Japanese with Ubuntu 14.04 Trusty Tahr suggest the following workaround:
I was not able to add the Anthy text entry in Ubuntu 14.04 using Gnome Fallback. Then I switch to Gnome Shell and I add it. Finally I switch back to Gnome Fallback and I can use it now.
I encountered the same problem after upgrading from 13.10 to 14.04: The Text Entry dialog in gnome-control-center does not list Japanese (Anthy), only plain Japanese and several others. Using the former, Super + Space (which previously was Ctrl + Space in 13.10), only toggles between the default keyboard layout (German for me) and English, but Anthy doesn't enable, even though it can be selected in the language selector applet (and then works and can be disabled once via keyboard, too).
For another user of that system, the toggling was still working, so I compared the configuration using dconf-editor. Under org.gnome.desktop.input-sources, I changed [('xkb', 'de'), ('xkb', 'ja')] to , [('xkb', 'de'), ('ibus', 'anthy')], logged out and in, and it worked!
