As others have already pointed out the answer is known as the Clebsch-Gordan formula. This reads as follows. W.l.o.g we may assume that $i\ge j$. The multiplicity of $\rho_k$ in the tensor product $\rho_i\rho_j$ is equal to 1, if $k$ is one of the integers in the arithmetic progression $i+j$, $i+j-2$, $i+j-4,\ldots$, $i-j+2$, $i-j$, and is zero otherwise.
One of the simplest argument showing this depends on the fact that all f.d. representations of $SU(2)$ split into a direct sum of 1-dimensional representations of the subgroup of diagonal matrices. These are classified by integers $\ell$ (aka weights) and correspond to the 1-dimensional representation, where the matrix $\pmatrix{e^{ix}&0\cr0&e^{-ix}\cr}$ acts with eigenvalue $e^{i\ell x}$ for any real $x$. In the representation $\rho_k$ of bivariate homogeneous polynomials of degree $k$ the monomial $x_1^{k-t}x_2^t$ has weight $\ell=k-2t$. Here $t$ ranges over the integers in the interval $0\le t\le k$, so the weights $k, k-2,\ldots,-k+2,-k$ appear in $\rho_k$ each with multiplicity one.
When we study the tensor product of two representations such $\rho_i\rho_j$ the weights are added up in the sense that if $u\in\rho_i$ has weight $s$ and $v\in\rho_j$ has weight $t$, then the elementary tensor $u\otimes v\in \rho_i\rho_j$ has weight $s+t$. Keeping a tally of the multiplicities of the weights of the tensor products of monomials coming from the two representations then quickly leads to the Clebsch-Gordan formula.
Students of quantum mechanics often encounter this calculation when quantum mechanical addition of angular momenta is explained.