Total angular momentum operators

Hi, we're really bad at quantum (and very confused) (although I hope
this is a valid question that actually means something)

Right, to recap the basics we (hope) we _do_ understand:

A particle has:
l, found by acting l^2 on the wavefunction (giving l(l+1)h^2)
m_l, found by acting l_z on the wavefunction (etc)
s, by acting s^2 on (the magic matrix part of) the wavefunction
m_z, ditto with s_z.
In some (bad analogy) sense l^2 is the "length" and m_l the "angle" of
the angular momentum.

Right, now here comes the confusion-causing part.
j is continually (and annoyingly) defined as l+s (which because they
are "vectors" isnt just a simple sum)
and we're told j has values allowed by the "crazy triangle rule"
(j=|l-s|...l+s)
the question is: seeing as a particle can be in a definite |j,m_j>
eigenstate how are these values actually encoded in the wavefunction?
(ie what is the actual explicit differential operator/matrix
representation of j^2?)

NB, please no answers based on l.s which are annoying and unhelpful.
.