$$I_{nm}=\frac{1}{\pi}\int_0^\pi \left(\frac{\sin nx}{\sin x}\right)^m dx$$

As stated in the OP, $I_{nm}=0$ for $n$ even and $m$ odd.
Otherwise, the nonzero $I_{nm}$ is the *central multinomial coefficient* $c_{nm}$, the largest coefficient of $(1+x+x^2\cdots x^{n-1})^m$.

A few examples:
$$I_{44}=44\;\; \text{and}\;\; (1 + x + x^2 + x^3)^4=1 + \cdots + 40 x^5 + \mathbf{44}\, x^6 + 40 x^7 +
\cdots + x^{12}$$
$$I_{34}=19\;\; \text{and}\;\; (1 + x + x^2 )^4=1 + \cdots + 16 x^3 + \mathbf{19}\, x^4 + 16 x^5 +
\cdots + x^{8}$$
$$I_{33}=7\;\; \text{and}\;\; (1 + x + x^2 )^3=1 + 3 x + 6 x^2 + \mathbf{7}\, x^3 + 6 x^4 + 3 x^5 + x^6$$

For small $n$ the central multinomial coefficients are listed on OEIS, for example, $I_{6m}$ is given for $m=2,4,6,8,\ldots$ by A063419:

_{
$$ 6, 146, 4332, 135954, 4395456, 144840476, 4836766584, 163112472594, 5542414273884, 189456975899496, 6507792553644256, \ldots$$
}

An efficient method to calculate these coefficients is described in On a link between Dirichlet kernels and central multinomial coefficients.

**Update 1**, following the conjecture added in a comment, that $I_{nm}=P_m(n)$ is a polynomial in $n$ of degree $m-1$; I have no proof, but for the record, here is the evidence for small $m$:

_{
$$P_1(n)=1$$
$$P_2(n)=n$$
$$P_{3}(n)=\frac{1}{4} \left(3 n^2+1\right)$$
$$P_4(n)=\frac{1}{3} n\left(2 n^2+1\right)$$
$$P_{5}(n)=\frac{1}{192} \left(115 n^4+50 n^2+27\right)$$
$$P_6(n)=\frac{1}{20} n \left(11 n^4+5 n^2+4\right)$$
$$P_7(n)=\frac{7 \left(841 n^6+385 n^4+259n^2\right)+1125}{11520}$$
$$P_8(n)=\frac{1}{315} n \left(151 n^6+70 n^4+49 n^2+45\right)$$
$$P_9(n)=\frac{259723 n^8+121380 n^6+84882 n^4+64580 n^2+42875}{573440}$$
$$P_{10}(n)=\frac{1}{36288}n \left(15619 n^8+7350 n^6+5187 n^4+4100 n^2+4032\right)$$
}

For fixed *even* $m$, these formulas represent closed form expressions for the central multinomial coefficients $c_{nm}$. On OEIS I find that such polynomial expressions are noted by R.H. Hardin (see A071816 and A005900), but without a reference to a general proof.

For fixed *odd* $m$, only the $c_{nm}$'s with odd $n$ follow. I have checked that there is no degree $m-1$ polynomial expressions for the full set of central multinomial coefficients with odd $m$.

**Update 2**
I have now located a proof in the literature, see https://mathoverflow.net/a/313854/11260