I haven't seen this structure before, but I'm extrapolating from things I know.
nanbon = なんぼん =何本 = how many [things that are counted with 本]
nanika = なにか=何か= something
If なんぼんか means "some number of things counted with 本", then なんぼんかのはなwould mean "some of the flowers" ... are paper? Is that the kami? It's really hard to tell with just hiragana.
But since this is just my best guess, I plan to follow this question in case a native speaker or someone who's studied longer has a better answer! ^^;