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The way I see it, there are two different (but related) ni's.
I will call them ni1 (roughly the one described by Adam) and
ni2 (roughly the one described by xod). They can be paraphrased
in terms of {du'u/ka} and {poi'i} respectively. They are
exactly parallel to the two competing uses of {jei}, as {du'u
xukau} and as truth value. (The {du'u xukau} use of {jei} is
mercifully dead these days, but given that it is in CLL it
could resurrect on us any time.)
The use of ni I'm most familiar with is ni1, but I believe xod
when he says many people use it as ni2. First, let's consider
ni1:
la djan frica la meris le ka ce'u dunda
John differs from Mary in their giving.
That could mean a lot of different things: they differ in
what they give, they differ in who they give to, they differ
in how many things they give, they differ in how many people
they give to, they differ in how often they give, etc, etc.
We have ways of being more precise:
la djan frica la meris le ka ce'u dunda makau
John differs from Mary in what they give.
la djan frica la meris le ka ce'u dunda fi makau
John differs from Mary in who they give to.
la djan frica la meris le ka ce'u dunda xokau da
John differs from Mary in how many things they give.
la djan frica la meris le ka ce'u dunda fi makau
John differs from Mary in how many they give to.
la djan frica la meris le ka ce'u xokauroi dunda
John differs from Mary in how many times they give.
Now, {ni1} is less specific than the last three, but more specific
than {ka} because it selects only the xokau-properties. So:
la djan frica la meris le ni ce'u dunda
John differs from Mary in how much they give.
Which could mean, in how many things they give, to how many people,
how many times, etc. We could in general say that
{ni1} = {ka sela'u makau}, if {sela'u} can pick any quantity
related to the situation.
Now {ni2} is not a du'u/ka, it is a namcu. Instead of being
a du'u containing a number, it extracts the number from the
situation decribed by the du'u. We have {poi'i} to extract
a participant in a relationship, but it only extracts sumti,
so we can use {mo'e ce'u} as the quantifier variable:
li ci ni le djan dunda
3 is how many/much John gives.
This could be:
li ci poi'i la djan dunda vei mo'e ce'u da
3 is the number of things that John gives
or:
li ci poi'i la djan dunda fi vei mo'e ce'u da
3 is the number of people John gives to.
or:
*li ci poi'i la djan vei mo'e ce'u roi dunda
3 is the number of times John gives.
(This one is ungrammatical, but I can't see a
grammatical way to put a sumti variable with roi)
or any other possible number extractable from the situation.
Conceivably, this number could even be a truth value:
li pimu poi'i la djan ja'a xi vei mo'e ce'u dunda
0.5 is the "ja'a xi"-value of "John gives".
so that we could say that {jei} is a special case of {ni}.
(Also in the case of ni1, we get the indirect question jei:
{du'u ja'a xi xokau}.
In summary: {ni} is vague as to which quantity of the event it
extracts. It has two versions:
ni1 = du'u sela'u xokau
ni2 = poi'i sela'u ce'u
mu'o mi'e xorxes
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