jmc

For $D767E89$ to be divisible by $9,$ we must have $$D+7+6+7+E+8+9=37+D+E$$ divisible by $9.$ Since $D$ and $E$ are each a single digit, we know each is between $0$ and $9.$ Therefore, $D+E$ is between $0$ and $18.$ Therefore, $37+D+E$ is between $37$ and $55.$ The numbers between $37$ and $55$ that are divisible by $9$ are $45$ and $54.$

If $37+D+E=45,$ then $D+E=8.$

If $37+D+E=54,$ then $D+E=17.$

Therefore, the possible values of $D+E$ are $8$ and $17.$ Our answer is then $8+17=\boxed{25}.$