  We introduce the transverse momentum dependent fragmenting jet function
(TMDFJF), which appears in factorization theorems for cross sections for jets
with an identified hadron. These are functions of $z$, the hadron's
longitudinal momentum fraction, and transverse momentum,
$\boldsymbol{\mathrm{p}}_{\perp}$, relative to the jet axis. In the framework
of Soft-Collinear Effective Theory (SCET) we derive the TMDFJF from both a
factorized SCET cross section and the TMD fragmentation function defined in the
literature. The TMDFJFs are factorized into distinct collinear and
soft-collinear modes by matching onto SCET$_+$. As TMD calculations contain
rapidity divergences, both the renormalization group (RG) and rapidity
renormalization group (RRG) must be used to provide resummed calculations with
next-to-leading-logarithm prime (NLL') accuracy. We apply our formalism to the
production of $J/\psi$ within jets initiated by gluons. In this case the TMDFJF
can be calculated in terms of NRQCD (Non-relativistic quantum chromodynamics)
fragmentation functions. We find that when the $J/\psi$ carries a significant
fraction of the jet energy, the $p_T$ and $z$ distributions differ for
different NRQCD production mechanisms. Another observable with discriminating
power is the average angle that the $J/\psi$ makes with the jet axis.
