This achievement is Story Based and impossible to miss, and can be done on any difficulty. However, here are some tips for the battle in case you have trouble on higher difficulties. For my current playthrough on Zealot the Ripper has proven indispensible, even without upgrades.
Recommended Inventory ::
Ripper + Ripper Blades
Plasma Cutter + Plasma Ammo
Pulse Rifle + Ammo
Any other weapon you favor or upgraded.
HOW TO GET THE RIPPER ::
In Chapter 5, you will go through a cryo morgue, and at the end of it you'll be forced to take a cargo lift down into a small room with a turning blue gravity generator. This is the same room with multiple stasis refill stations, a gravity console, and several kinesis arms hanging around the gravity generator that you have to pull down to progress in the story.
In this room is the RIPPER Schematic. Grab it, and buy it at the next store. By the time you get to Chapter 6 (the school) you should have a fair amount of Ripper Blades stocked up.
THE SCHOOL ::
A note before you get to the school, is that through this chapter (but not in the school itself) you will inevitably encounter Cysts, which are little round balls of flesh connected stationary to walls. They shoot fleshy tentacle mines that will explode if you get close. It is recommended to use Kinesis on them at least once and fire them back for the Bouncing Betty achievement.
In the school itself you will encounter a new type of enemy, a glowing, crawling baby known as the Crawler. They will explode if they get close or you shoot their bellies, so take care to not blow them up if they are close to you, and for maximum effectiveness you can clear out a group by abusing splash damage.
Chaining Crawler explosions is the best way to survive with minimal ammunition expenditures, but bear in mind that if you can kill them without blowing them up by shooting their heads or tiny limbs, you get credit towards The Nanny achievement.
This is why I recommend bringing a precision instrument like the trusty plasma cutter or the pulse rifle. Using Plasma Cutter's horizontal fire and aiming at a downward angle towards their heads and legs is relatively easy even without stasis due to the Crawler's fragility. The Pulse Rifle truly shines here because of how ludicrously efficient your bullet to kill ratio can be.
There are not enough Crawlers in Chapter 6 alone to get The Nanny in its entirety, so don't stress when you beat Chapter 6 without earning it. Also of note is that you can kill Crawlers by removing their limbs to get credit towards The Nanny, and their torso will usually remain to abuse with an additional shot or Kinesis to help control a horde.
Towards the end you will enter a gymnasium type area with a stage, and bleachers. As soon as you enter here, you will be engaged by the Pack, which is the small children who just mindlessly swarm you in large numbers. When you enter, you can turn right and back yourself into a corner and face the Pack and let loose with the Primary Fire of the Ripper. This will cause the Pack to run directly at you like mindless fools, and die at the hands of your Ripper. This and the battle afterwards on the Stage is a good opportunity to get the achievement to kill 4 enemies with 1 Ripper Blade.
You will be forced to go behind the Bleachers to progress and doing so will spawn your first advanced (black) necromorph, so be sure to have some Stasis charge ready and let loose on his limbs with the Ripper or another appropriate gun.
After this you will go onto the Stage, and backstage you will need to activate a switch to progress. Doing so will cause a massive horde of Crawlers and Pack to spawn, so if you get backed into a corner again with the Ripper you can fight them off. However, note that by backing yourself into this corner you need to be wary of Crawlers.
The way I did this on Zealot was holding Primary Fire to kill all Pack and as soon as one of the slow Crawlers was in range I would use Alt Fire of the Ripper to detonate it before it got to me. This clearing effect mixed with liberal applications of stasis will ensure that you get through this area with absolutely no or minimal damage taken.
Afterwards you will leave through the door you exposed into a new room/area and its a smooth sail to the exit - ACHIEVEMENT UNLOCKED!