The Kermadecs are certainly interesting.
Looking at the New Zealand EEZ map, I think we can be sure the
southern boundary of the Kermadec EEZ is based on l'Esperance Rock,
despite the dodginess of such a claim - an island less than a km
across called "Rock" doesn't sound like it meets the UN criterion of
habitability. (But hey, my native land claims Rockall.) But the
southernmost island in the chain is Curtis, a good 100km farther
north, and its EEZ would fall noticeably short of the NZ EEZ.
So the southern edge of the Kermadec EEZ can be scribed 200nm south
of 31d26mS 178d54mW (these coords and others from the GEOnet name
server).
I've surveyed the northern NZ coast around Bay of Plenty, and found
what seems to be the two points closest to l'Esperance: to the west,
Needles Point on Great Barrier Island (36d02mS 175d24mE); to the
east, Matakawa Point (37d34mS 178d20mE). Coordinates to one-minute
accuracy have an error of around 2km, so for my calculations I've
assumed that Esperance is 30s of arc farther N and W, and that the NZ
points are 30s of arc S and E - therefore pushing them to the maximum
possible distance apart compatible with the given coordinates.
Approximating the local ellipsoid with a sphere that has the same
radius as the mean radius of the Earth for the three points
described, I get the following:
l'Esperance to Needles Point = 729.9km
l'Esperance to Matakawa Point = 736.6km
One nautical mile is 1852m, so 400nm = 740.8nm
So even making conservative assumptions, I get a slim EEZ overlap at
both locations, leaving the Bay of Plenty - Kermadec gap as a true
exclave of the high seas.
I've given all the tedious info because I'd love someone else to
check my figures; the situation is obviously marginal, but I can't
see how the distances can be any greater by even a kilometre unless
the coordinates are wrong or my sums are amiss.

Grant