This will probably not work. The 7806 is only good for up to 1.5A, and that only when power dissipation it's handled adequately.
The drill motor might draw less than 1.5A when free-running, but when it's actually generating torque turning a screw, I suspect it pulls much more.
Also, dropping 6V @ 1.5A means 9W dissipation in the regulator. Your little heat sink is likely not up to the task.
Finally, you should have some capacitance in the input and output of that regulator.
On the other hand, the 78xx are pretty robust little guys and are more likely to shut down when unhappy than just break, so you might get away with light intermittent use.