This is the hardest one of these 'less than 3 hits' to do, due to the way the boss runs away from you.
The boss stage is comprised of two 'areas' - the part leading up to the lever, aka "Part A", and the part where you're facing off the boss and his minions, aka "Part B". The general formula is that you start in Part A, flip the lever, and during Part B you hit the boss once, and are taken back to Part A to repeat the process.
During Part A, there will be a trio of underground missiles which will follow you, and you'll need to lead at least one of them to the shield protecting the lever to be taken to Part B.
Towards the end of Part A, where there's a second trio of missiles, one of those missiles will grant you a drill Wisp.
After obtaining the drill Wisp, navigate to Part B, wait for the boss to come out of the background, then change into Drill. You'll then need to do your best to chase him and hit him within 5 seconds. If you take longer than 5 seconds, the boss will pick up speed, making him impossible to chase, forcing you to hit him normally (which in turn will void the 3 hits achievement, therefore forcing you to restart the stage).
This may take you a few goes, but still very doable. You just need to pull it off twice in a row essentially. Good luck!