This exercise is aimed at showing that zero-padding in the time domain interpolates the frequency domain. Since this is homework, I'll give you the beginning of the solution.
Just like you did, start with:
\begin{align} Y[k] &= \sum_{k=0}^{N-1}x[n]e^{-j2\pi k\frac{n}{2N}}\\ &= \sum_{k=0}^{N-1}x[n]e^{j2\pi k\frac{n}{2N}}e^{-j2\pi k\frac{n}{N}}\\ &= \mathcal{F}\left\{x[n]e^{j2\pi k\frac{n}{2N}}\right\} \end{align} where $\mathcal{F}$ denotes the DFT operator.
Next, we know that multiplication in the time domain is convolution in the frequency domain, so: $$Y[k] = X[k] * \mathcal{F}\left\{e^{j\pi k\frac{n}{N}}\right\}$$
All that's left is to compute $\mathcal{F}\left\{e^{j\pi k\frac{n}{N}}\right\}$
Can you take it from there?