For $\mu$ a weight, let $||\mu||_1$ denote the one-norm of $\mu$ (the sum of the absolute values of its entries) and let $Z(\mu)$ be the number of zero coordinates of $\mu$. Let $k\geq0$ and $1\leq p\leq n$. Write $r(\mu)=(k+p-||\mu||_1)/2$. If $r(\mu)$ is a non-negative integer, then \begin{align*} m_{\pi_{\Lambda_{k,p}}}(\mu) &= \sum_{j=1}^{p} (-1)^{j-1} \sum_{t=0}^{\lfloor\frac{p-j}{2}\rfloor} \binom{n-p+j+2t}{t} \sum_{\beta=0}^{p-j-2t} 2^{p-j-2t-\beta} \binom{n-Z(\mu)}{\beta} \binom{Z(\mu)}{p-j-2t-\beta} \\ &\quad \sum_{\alpha=0}^\beta \binom{\beta}{\alpha} \sum_{i=0}^{j-1} \binom{r(\mu)-i-p+\alpha+t+j+n-2}{n-2}, \end{align*} and $m_{\pi_{k,p}}(\mu)=0$ otherwise.
This is Theorem 4.1 in this article, where there are also analogous weight multiplicity formulas for the rest of the classical complex Lie algebras.