If $\frac{a}{a-4} + \frac{2}{a} = \frac{a^2+2a-8}{a(a-4)}$ is an integer, then $a(a-4)$ divides $a^2+2a-8$. So, $a$ divides $a^2+2a-8$, and $a$ divides $8$. All integer divisors of $8$ different from $4$ are $1$, $2$, $8$, $-1$, $-2$, $-4$ and $-8$. We check all seven cases and see that the value of the expression is an integer only when $a = 2$ or $a = -4$.