Whats the amperage of the drill battery? If you can determine that then you can see how long it will run (I am guessing not that long but to be honest I have never tried it).
Are you going to try to recharge it through the laptops charging port or are you going to unhook the battery each time you need to charge it (to drop it into the drill charger).
If this works (for a reasonable amount of time) it would be great. You can get the blue "noname" drills for just a few bucks (usually under $20), strip out the battery for use with the laptop, and pull out the motor for robotics (or other things).