(I'm aware the thread is 2 weeks old, so stuff could have changed in between (which I'm not aware of btw).
Go to the Shop => Material exchange => Materials.
You can use the fragments to get yourself a "Gale Gem Idea Regalia" (5 fragments gives you one gem).
OR... If you got 6 fragments you can combine it with a 'Howling Idea Core' (you'll need 3) which gives you the Prison of Wind Idea Core. This item is required to start the Ragnarokk Wind raid quest (whatever it's called).
Of course this leads up to another question (or it should ) => "What does the Regalia do?". I'm not sure but I think this item can be used to level break your Kamihime but... Not sure. Still hope this can help.