hello
kobe airport http://www.japan-guide.com/e/e2432.html only offers domestic flights to tokyo with jal airlines and nippon airways.
you could always book a multi leg flight on the webb, london - tokyo - new zealand - london and arrange a domestic rtn flight from tokyo to kobe.
do you have any spefic dates planned and where do you intend to go in new zealand? also what is your local airport in the UK. i could have a quick look and see what the webb offers you and get back to you within 48hrs. i may or maybe not find something for you, but willing to give it a go.
i would have to have a rough idea on how long you wish to stopover in kobe, new zealand etc....
kind regards
jim