Here's a different approach. Since this is homework, I only give an informal description with a few details missing; see if you can make this rigorous.
Let $S = \{ y_1, \ldots, y_n \}$ and $S_{-i} = S \smallsetminus \{ y_i \}$. As a first step, we find $n$ dual vectors $z_1, z_2, \ldots, z_n$ such that $ \langle z_i, y_j \rangle_{\ast} = \delta_{ij}. $ In fact, it is easy to write down what $z_i$ should be. Note that $\operatorname{Span}S_{-i}$ is an $(n-1)$-dimensional subspace orthogonal to $z_i$; so take the unit vector in the $1$-dimensional subspace $(\operatorname{Span} S_{-i} )^{\top}$ and normalize it suitably to make the inner product with $y_i$ equal to $1$.
Once we get the vectors $z_i$, note that $z_i = x_i^{\ast}$ for some $x_i \in V$.
Approach 2. In fact, this problem is trivial once you realise that $(V^*)^* = V$. Given the basis $\{ y_1, y_2, \ldots, y_n \}$ for $V^*$, find the dual basis $B$ for $(V^\ast)^\ast = V$. Then you should be able to show that $B$ is the desired basis.