A natural question which was likely studied in academic literature (even though Kelly is not particularly popular among portfolio managers). I guess you could generalize Eq. (6.87) of this book. If $f(R)$ is the joint probability density of the returns of your $N$ assets (so $R$ is $N$-dimensional vector), for example Gaussian,
$$
f(R)\propto\exp(-(1/2)\sum_{ij}C^{-1}_{ij}(R_i-\mu_i)(R_j-\mu_j)),
$$
and $x$ is an $N$-dimensional vector of your bets, the Kelly log utility is
$$
U(x)=\int\log(1 + x\cdot R)f(R)dR.
$$
The utility is maximized when $\partial U/\partial x_i=0$, or
$$
0 = \int\frac{R_i}{1+x\cdot R}f(R)dR.
$$
I don't know if this can be solved in a closed form, but maximizing $U(x)$ numerically seems doable. Note that in the classical Kelly case you need to consider only the slab are $0\le\sum_ix_i\le1$, but if you are long/short and levered, the constraint is lifted.