because it's great
tl;dr Sunshine Academy is a hidden minigame in Karamari Hospital. I'm not gonna spend this guide trying to explain what the purpose may be or anything like that, just guiding people to it and letting them get their own experience for it.

First of all, if you haven't beat Karamari Hospital, go do so.

Done? Alrighty then.
Remember that fun basement area? The one you had to open up in the offices, wait 12 years for the door to open, then go down more stairs than SCP-087 has? The game is down there.

Just as you enter the basement, take the first door on the left. Go down the hallway and travel left again, taking the door there. This brings you to a room, where Sunshine Academy sits on the table.
To the left. Nothing but left to find it. Even the waifu is looking to the left.

Well, play it. That's about it.

Some stuff may change each time you play the game. If you want a full and pure experience of your own, quit reading here. not that there's much to spoil anyway

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There's two total endings, a bad one and a good one. Which one is which is up for your personal interpretation so I won't assign either one a name.

The first ending is impossible to not get. It's the default ending, which you will get on your first few playthroughs. You know you got this ending when Ume's looking like she's having an average Monday and you're stuck on an infinite loop of cryptic and Spooky™ messages.
The second ending, however, takes a bit of work. First of all, you gotta get that un-fun ending a few times. Once you pull that off, you'll notice some strange dialogue options from the get-go. To get the second ending, you need to make sure everything you say sets off a red flag. Say weird, creepy, and disturbing stuff until Ume reports your strange behavior and you get transferred to another school, letting her escape you.
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That's about it, actually. Nothing else to do but play the game.
Now, as for lore and what-not, or whatever ties to the main game this one has, is up for us to figure out. So go down there, play with your newfound waifu a bunch, and cook up some over-the-top theories. Together, we can put way too much time and effort into this.

The two lines below are the two achievements you get from Sunshine Academy.
LOVE FAILED! - Get the secret ending of Sunshine Academy.
Backseat Gamer - Die to The Patient while playing Sunshine Academy.
You get this by initiating the chase with Monster 4/The Patient (the hanging guy on the other end of the basement) and then run to the game, launch it, and die.