Help evaluating a trace identity of $SU(2)$ generators

In solving a problem I came across the need to evaluate the following: lets say that $J_pm$, $J_z$ for an $N$ dimensional representation of $SU(2)$ and choose a $j<N$. In other words, the set of $J_i$‘s are three $Ntimes N$ matrices satisfying the commutation relations for $SU(2)$. Let $Tr$ denote the trace operation on the $N$ dimensional representation. The quantity I’m interested in evaluating is the ratio of traces:

begin{equation}
f(j,k,N)=frac{Tr[J_+^j J_3^k J_-^j]}{Tr[J_+^j J_-^j]}
end{equation}
For $k=1,2,3$. For $k=1$ I can take a trace of a commutator $[J_+J_z,J_-]$ which gives me that $f(j,1,N)=(-j/2)$. But I can’t seem to be able to do the same for higher values of $k$.