First things first you need to be playing a New Game +, e.g. you've completed the game at least once before on that save file. Secondly it's basically a matter of having high mental health; not just Green ending mental health, but mental health so high it's basically impossible to get even on a normal high mental health play through, so you need to follow special steps to get it that high (you need an S* rank at the ending for this).
Do the following to get your mental health high enough for this ending:
- Either play a pacifist play through or just avoid killing too many monsters. You don't have to, but it will greatly improve your mental health and require lest mental health grinding later on
- Eat good food, sleep when you need to, general high MH (mental health) stuff. You must do this early on because if your MH is high he gives you items that help you raise your MH even more later on.
- As per above, talk to The Director at every opportunity, he's instrumental to this. He gives several items for cooking better food.
- Adopt the cat by giving it all 5 cans of cat food (give the Director cat comics to get all 5 cans, they're in The City).
Attempt to give your Sleepy Cat to all human characters
- This includes Kenny, Bendizo, Chie, The Director, Hank and The Man In The Box (take a green pill), and also to the cat after giving it 5 tins of cat food.
- After getting all items from each Sleepy Cat receiver, you should be able to make an Ultimate Burger. Combine Bun with Patty, then add lettuce, then sliced cheese, then add pickles. He will declare it as the Ultimate Cheeseburger when you're done. Do NOT eat the last of any of these items before getting all other parts. You only get one of several.
- The burger is also good for a trophy, one very hard to get by accident.
- Giving sleepy cat to everyone is also a requirement for the Yellow Ending, so you might as well do it now even if you get high enough mental health otherwise.
Save your Cola/Mexican Cola items; after getting the cat key you can access a special room inside the hospital with a liquid nitrogen can. Use your colas on this canister to chill them; this gives two trophies on PS (one for each kind of cola) and greatly improves your mental health, which may push you over a tipping point if your MH is too low when you enter the hospital
Once in the hospital you should be able to open the first door from the left, the one You usually says he's too scared to open. If not, try freezing your colas as per above; if that's not enough you'll need to return to an earlier save and grind more mental health. Try talking to your cat a bunch, I believe that helps.
After entering the corridor in the hospital...
You will find Her, zombified. She'll crawl along the floor and will violently react to any attempts to harm Her. Do NOT shoot her, put down flares or rotten meat. You will be able to hide in several spots along the way, hide until Her passes by you then exit hiding and slip into the next hiding spot to the left. The last hiding spot is swarming with Thin Men but you should be fine if you hide carefully. If you fire a shot or Thin Men notice you, then you lose the ability to hide and will have to exit the room to the right and try again, or Her kills you.
After exiting the corridor to the left you'll get the White/best ending.
In the ending you find that Her seems to be dying, and You continues on without Her. Sort of a Best ending compared to the Blue one and it seems to wrap up Her's significance to the story more.