Some preliminary definitions. On page 342 of Lee's Smooth Manifold he concludes that $\hat{g}(\operatorname{grad} f)(X) = Xf$ where $\hat{g}$ is the isomorphism between $TM \to T^*M$, the tangent bundle and co-tangent bundle.
Now according to Lee, $\hat{g}^{-1}(\omega) = g^{ij}\omega_j \frac{\partial }{\partial x^j}$
But what I found is that,
\begin{align} \hat{g}(\hat{g}^{-1}(df))(X) &=\hat{g}(g^{ij}\frac{\partial f}{\partial x^j} \frac{\partial }{\partial x^j})(X)\\ &=g_{ij}g^{ij}\frac{\partial f}{\partial x^j}\frac{\partial }{\partial x^j} X^i\\ &=\frac{\partial f}{\partial x^j}\frac{\partial }{\partial x^j} X^i \\ &= X^i \frac{\partial f}{\partial x^j} \frac{\partial }{\partial x^j} \text{because g is symmetric, so we can bring X to the front} \end{align}
But $Xf = X^i(x)\frac{\partial f}{\partial x^j}$. So there is an extra basis term coming out of my reduction. What's going on?
First of all you have mixed the indices a bit. In the first line we should have
$$\hat{g}(\hat{g}^{-1}(df))(X) =\hat{g}\left(g^{ij}\frac{\partial f}{\partial x^j}\frac{\partial }{\partial x^\color{red}{i}}\right) X$$
For the second line you shouldn't keep $Y=g^{ij}\frac{\partial f}{\partial x^j}\frac{\partial }{\partial x^{i}}$ when you evaluate $g$. Instead you should just take the $k$-th component. This is explicitly stated at the beggining of p. 342. Indeed $\hat{g}(Y)(X)$ should produce a number, not a vector like in your case. Thus we would get:
$$\hat{g}(\hat{g}^{-1}(df))(X) =\hat{g}\left(g^{ij}\frac{\partial f}{\partial x^j}\frac{\partial }{\partial x^i}\right) X = g_{ik} X^i g^{kj}\frac{\partial f}{\partial x^j} = \delta^j_i X^i \frac{\partial f}{\partial x^j} = X^i\frac{\partial f}{\partial x^i} = Xf$$
Note that $Xf = X^i\frac{\partial f}{\partial x^\color{red}{i}}$, not $Xf = X^i\frac{\partial f}{\partial x^j}$, as you have mentioned in the last line.
REMARK: The reason why I take the $k$-th component is because we have already used $j$ for the coordinate representation of $\hat{g}^{-1}(df)$. Thus $g(X)(Y) = g_{ij}X^iY^k$. In fact I believe that this is the reason behind your confusion in the probem.