I think I agree that quidquid cannot normally be split up; but is that reason enough to say it must be one word? The particle -que cannot normally be split up either, but it is split up occasionally in poetry, if I remember correctly. I think what constitutes a word and what doesn't is ultimately an unreclaimable quagmire, though in this case I'd certainly prefer quidquid over quid quid too.
If anyone can do non-wrong Latin, I could use a translation of the following for HPMOR. The original is supposed to be circa 1200.
No rescuer hath the rescuer.
No Lord hath the champion,
no mother and no father,
only nothingness above.