Two Point Campus

You may encounter a glitch in this achievement that will prevent you from unlocking the achievement even after making sure every single illness was done. My tracker didn't update until I started doing specific illness' researches that were meant to boost treatment rate for a specific illness. The tracker started going up by 10% (which mathematically doesn't match the 24 illnesses in denominator). If you're stuck and not sure what to do to make it work, give it a go with the researches. One thing to note is that I got the illnesses on PC first but then I completed the achievement on console.

However, getting into how it's supposed to be done. There are 24 illnesses in this game:
Psychiatry (5):
Mock Star
Mime Crisis
Space Caged
Night Fever
Reptile Dysfunction

Ward (6):
Flappy Hamper
Limpette
Broken Face
Boneless Thighs
Pine Dented
Piste Off

Head Clinic (4):
Lightheadedness
Screwheadedness
Rootheadedness
Metalheadedness

Injection Room (3):
Greenvious
Cheese Bored
Leopard Skin

Thumping Therapy (3):
Blank Look
Block Head
Woodworms

Surgery (3):
Bodily Druids
Heart Throb
Brain Farts

You should get the achievement around the time you get 1 star in Pointy Peak.