Could you please do the following:

1. Eject your DVD/CD player and remove the Divinity 2 disc from the drive.
2. Start Flames of Vengeance (particularly Flames of Vengeance, NOT Ego Draconis)
3. Fill in the activation code we gave you and click Activate

Does this work?

And if you quit the game, and then try to start it again, does it ask for the activation key again?

If you have installed the digital version of Flames of Vengeance, like you have, the disc is not required anymore.