$\begin{align}
\mathbb{E}[X | Y = 5]
& = \sum_{n = 1}^{\infty} n \, \mathbb{P}(X = n | Y = 5)
= \sum_{n = 1}^{\infty} n \cdot \frac{\mathbb{P}(X = n, Y = 5)}{\mathbb{P}(Y = 5)} \\
& = \sum_{n = 1}^{4} n \cdot \frac{\mathbb{P}(X = n, Y = 5)}{\mathbb{P}(Y = 5)} + \sum_{n = 6}^{\infty} n \cdot \frac{\mathbb{P}(X = n, Y = 5)}{\mathbb{P}(Y = 5)} \\
& = \sum_{n = 1}^{4} n \cdot \frac{\left(\frac{4}{6}\right)^{n - 1} \cdot \frac{1}{6} \cdot \left(\frac{5}{6}\right)^{4 - n} \cdot \frac{1}{6}}{\left(\frac{5}{6}\right)^4 \cdot \frac{1}{6}} + \sum_{n = 6}^{\infty} n \cdot \frac{\left(\frac{4}{6}\right)^4 \cdot \frac{1}{6} \cdot \left(\frac{5}{6}\right)^{n - 6} \cdot \frac{1}{6}}{\left(\frac{5}{6}\right)^4 \cdot \frac{1}{6}} \\
& = \frac{3637}{625}
\end{align}$