I'm over a year late to answer this question, but here goes. I think you must have misapplied some identities or got lost in the weeds during your derivation. This has to do with the torsion-freeness of the connection. We have, using the formulas for the covariant derivative of a rank 2 tensor:
\begin{align*}
D_rD_sV_n=&D_r(\partial_s V_n-\Gamma^u_{sn}V_u)\\
&=\partial_r \partial_s V_n-(\partial_r \Gamma^u_{sn})V_u-\Gamma^u_{sn}\partial_r V_u\\
&-\Gamma^a_{rs}(\partial_a V_n-\Gamma^u_{an}V_u)\\
&-\Gamma^a_{rn}(\partial_s V_a-\Gamma^u_{sa}V_u)
\end{align*}
We want to antisymmetrize with respect to r and s, and I want to use bracket notation $V_{[a} U_{b]}\equiv \frac{1}{2}(V_a U_b-V_b U_a)$ for brevity. To do this without ambiguity, we should try to rearrange all our indices to put r and s next to each other. Using the symmetry of the Christoffel symbol in the lower two indices (a consequence of torsion freeness)...
\begin{align*}
D_rD_sV_n=&-(\partial_r \Gamma^u_{sn})V_u-\Gamma^u_{ns}\partial_r V_u\\
&-\Gamma^a_{nr}\partial_s V_a+\Gamma^a_{nr}\Gamma^u_{sa}V_u\\
&+\text{symmetric stuff}
\end{align*}
Then we have:
$$
D_{[r}D_{s]}V_n=-(\partial_{[r} \Gamma^u_{s]n})V_u-\Gamma^u_{n[s}\partial_{r]} V_u-\Gamma^a_{n[r}\partial_{s]} V_a+\Gamma^a_{n[r}\Gamma^u_{s]a}V_u$$
The first and last terms give the Riemann tensor up to factors of two. The middle two terms add up to zero because by definition, $\Gamma^u_{n[s}\partial_{r]}=-\Gamma^u_{n[r}\partial_{s]}$.
What did you do wrong?
You might have forgot to properly distribute the partial derivative across $\partial_r(\Gamma^u_{sn}V_u)$. If you work with the convention that $\partial_r$ acts on everything to the right of it, you have to be very careful about it. If you find yourself making mistakes or overusing parentheses, you can use the symbol $\overrightarrow\partial$ to mean "acts on everything to the right" and the symbol $\partial$ to mean "acts on the thing immediately to the right". So as an operator equation $\overrightarrow\partial_x f=\partial_xf+f\overrightarrow\partial_x$.
You also might have misapplied the covariant derivative of a 2-tensor. We have:
$$D_a (V_{bc})=\partial_a V_{bc}-\Gamma^u_{ab}V_{uc}-\Gamma^u_{ac}V_{bu}$$
There are also plenty of other bookkeeping / index mistakes that could be made here.
Other References
pg. 82 / equation 3.66 of Sean Carroll's lecture notes covers this, making no assumptions about torsion.
