Problem proving a tensor identity

I am asked to prove
$$nabla cdot (T cdot v) = T : nabla v + vcdot (nabla cdot T)$$
Where $T$ is a order-2 tensor and $v$ is vector, in an orthogonal basis. Let $delta _{ij}$ denote the Kronecker delta.

My proof:
$$T = T_{ij} e_i e_j, v = v_k e_k$$
where $e_i$ are unit vectors in an orthogonal basis.
So,
$$T cdot v = T_{ij}v_k e_i e_j cdot e_k = T_{ij}v_k e_i delta _{jk} = T_{im}v_me_i
implies nabla cdot (T cdot v) = frac{partial (T_{im}v_m e_i)}{partial x_n}cdot e_n = frac{partial (T_{im}v_m)}{partial x_n} delta_{in}=frac{partial (T_{pm}v_m)}{partial x_p} = frac{partial (T_{pq}v_q)}{partial x_p} $$

Moving on to the RHS:
$$T:nabla v = T_{ij}e_i e_j : e_nfrac{partial (v_m e_m)}{partial x_n} = T_{ij}frac{partial v_m}{x_n}e_ie_j:e_ne_m = T_{ij}V_{mn}delta_{jn}delta_{im}=T_{qp}V_{qp}=T_{qp}frac{partial v_q}{partial x_p}$$
The second term of the RHS:
$$vcdot(nabla cdot T)=(v_ie_i)cdotleft(e_j cdot frac{partial}{partial x_j}(T_{kl} e_k e_l)right)=(v_ie_i)cdotleft(frac{partial T_{kl}}{partial x_j} e_ldelta _{jk}right) = (v_ie_i)cdotleft(frac{partial T_{jl}}{partial x_j} e_lright) = v_ifrac{partial T_{jl}}{partial x_j} delta _{il}=v_qfrac{partial T_{pq}}{partial x_p}$$
Where I replaced $m$ with $p$, and contracted $i$,$k$ with the index $p$ in the last statement.

As you can see, $$v_qfrac{partial T_{pq}}{partial x_{p}} + T_{qp}frac{partial v_{q}}{partial x_{p}} neq frac{partial T_{pq}v_q}{x_p}$$

What am I doing wrong here? Any advice you have would be appreciated!

Initially, I had made a mistake wrt to the divergence operator, but I realize now that the error is in the first term, not the second term.