Structure coefficients of a coframe

Nothing wrong with your calculation. In the book, the author distinguish $\partial_i$ with $\frac{\partial}{\partial x^i}$, as follows : While $\frac{\partial}{\partial x^i}$ is the usual partial derivatives w.r.t. $x^i$ coordinate, the symbol $\partial_i$ is used as the components of exterior derivatives of a function when we express $df$ in coframe $\theta^i$, she called $\partial_i$ as "The Pfaffian Derivatives". That is (in the same page, before equation $I.4.2$) $$ df \equiv \partial_if \, \theta^i. $$ So now you have a choice to find $df$. First as you do $df = \frac{\partial f}{\partial x^i} dx^i$, or you can use $df = \partial_if \, \theta^i$. So now $$ d\theta^i = d(a^i_j dx^j) = da^i_j \wedge dx^j = (\partial_ka^i_j \, \theta^k \wedge dx^j) = \partial_ka^i_j \, \theta^k \wedge (A^j_l \theta^l) = A^j_l \partial_ka^i_j \, \theta^k \wedge \theta^l. $$ As @JohnMa said, the expression $d\theta^i = -\frac{1}{2}C^i_{kl} \, \theta^k \wedge \theta^l $ is in the antisymmetrized form of $d\theta^i$. So we take the antisymmetrization for $d\theta^i = A^j_l \partial_ka^i_j \, \theta^k \wedge \theta^l$ as \begin{align} d\theta^i &= \frac{1}{2}(A^j_l \partial_ka^i_j - A^j_k \partial_la^i_j) \theta^k \wedge \theta^l + \frac{1}{2}(A^j_k \partial_la^i_j-A^j_l \partial_ka^i_j ) \, \theta^l \wedge \theta^k \\ &= -\frac{1}{2}C^i_{kl} \, \theta^k \wedge \theta^l \end{align} So $$ -\frac{1}{2}C^i_{kl} = \frac{1}{2}(A^j_l \partial_ka^i_j - A^j_k \partial_la^i_j) \implies C^i_{kl} = A^j_k \partial_la^i_j - A^j_l \partial_ka^i_j $$ Note that this result $C^i_{hk}=A^j_h \partial_ka^i_j - A^j_k \partial_ha^i_j$ is different (reversed $h$ and $k$) with the one that given on the book, that is $C^i_{hk}=A^j_k \partial_ha^i_j - A^j_h \partial_ka^i_j$. If you're going through earlier computation in the book, about $d^2f$, you'll find that the eq. given in the book $C^i_{hk}=A^j_k \partial_ha^i_j - A^j_h \partial_ka^i_j$ is not match, which means that probably there are typos (indices $h,k$ are reversed) there. It should be $C^i_{hk}=A^j_h \partial_ka^i_j - A^j_k \partial_ha^i_j$.