\begin{align}
[H,\rho] & =\sum_{ikm}n_i\sigma_{i,jk}\delta_{lm}\rho_{kp,mq}-\rho_{jk,lm}n_i\sigma_{i,kp}\delta_{mq}\\
& = \sum_{ik}n_i\sigma_{i,jk}\delta_{ll}\rho_{kp,lq}-\rho_{jk,lq}n_i\sigma_{i,kp}\delta_{qq}\\
& = \sum_{ik}n_i\sigma_{i,jk}\rho_{kp,lq}-\rho_{jk,lq}n_i\sigma_{i,kp}\\
& = \sum_{ik}n_i\left( \sigma_{i,jk}\rho_{kp,lq}-\rho_{jk,lq}\sigma_{i,kp} \right)
\end{align}
Finally it makes sense! Now to see how to simplify further to show that all off diagonal entries in space 1 vanishes to complete the proof in index notation