In Skyloft, you found out that you need to find the missing propeller in order to point both windmills at the Light Tower.
Get a Cheap Fortune
After you reactivate Scrapper, the fortune teller will offer a one-time-only discount, where you can pay just 1 rupee to get your fortune told. It is not recommended to use this discount to ask about Treasure.
Buy New Things
You can now buy the Sacred Shield from Rupin. It isn't weak to elemental attacks like the Wooden Shield and Iron Shield, and it can repair itself over time, but it is somewhat flimsy. The potion lady has a new Stamina Potion that temporarily makes your stamina last longer. It is recommended to buy a Stamina Potion now, because it will be useful soon.
Talk to Dovos in the Bazaar at the cafeteria. He will tell you a rumor of a woman crying in the dormitory at night. Talk to Henya in the Knight Academy kitchen and she will say she hears a voice in the restroom at night. Sleep until night and then approach the door of the Knight Academy restroom. The person inside will ask for paper.
If you got a Stamina Potion at the Bazaar, go to Fledge's room at night and give him the Stamina Potion.
Deliver Cawlin's Letter
During the day, talk to Cawlin near the restroom of the Knight Academy. He gives you a love letter for Karane.
You can either give the letter to the person in the restroom (at night), or you can deliver it to Karane (during the day). You won't be able to change your choice later. You will still get the same reward, but the outcome will be different. You can also meet the person in the restroom but still choose to give the letter to Karane instead.
If you want to give the letter to Karane, you can find her in the library during the day. After you give her the letter, you can optionally go back and talk to Cawlin. Then go upstairs and talk to Pipit. Then go back downstairs and enter the library. After the cutscene, you get five Gratitude Crystals.
If you want to give the letter to the person in the restroom instead, sleep until night, then check on the restroom, then go in. Approach the toilet, talk to the hand, and give it the paper. Say you are sure. Then sleep until morning. Talk to Cawlin near the restroom, then go to Groose's room and talk to Cawlin again. Then sleep until night, and go into Groose's room and talk to Cawlin. You will get five Gratitude Crystals.
Clean Pipit and Mallara's House
Go to Pipit's house in eastern Skyloft. It is near the bridge and is a red house with a blue diamond over the door. Pipit's mother, Mallara, will ask you to clean the house. Use the Gust Bellows to blow away all of the dust and cobwebs on the floor, furniture, and windows. Don't worry about pointing the Gust Bellows at her; it won't hurt her and it is somewhat entertaining. When all the dust is gone, she will tell you that the job is done. Talk to her to get five Gratitude Crystals and 20 rupees.
Get Rewards from Batreaux
You might now have more than 30 Gratitude Crystals. If so, talk to him to get the Big Wallet. A chest will also appear. It contains the Cursed Medal, which greatly increases the rupees and treasure that you find, but it also prevents you from using anything else in your pouch (shields, potions, fairies, etc.) You can keep the Cursed Medal into the Item Check to be able to use your items again, but you won't get the rupee and treasure boost from the Cursed Medal while it is in Item Check.
Finish your Lumpy Pumpkin Tasks
Go into the Lumpy Pumpkin and talk to Pumm at the bar. He says that the next task needs to be done at night. Sleep in a bed on the first floor of the Lumpy Pumpkin.
Talk to Pumm at night and he will ask you to do a duet with Kina. The people in the audience will wave their arms in the direction that you should strum the harp. Try not to be hasty. Otherwise, you might end up moving when they don't move. There are some points at which they quickly change arm position that you need to watch out for. Try not to strum too broadly, or else you might strum when you don't mean to.
The background music will include more instruments if you are doing a good job. The audience will have a better reaction at the end if you did well. If your performance is not satisfactory, you will have to try again. If you did a good enough job, you will be done with all of the tasks. Pumm will give you a Piece of Heart. Sleep in a bed until morning and you can see the new chandelier.
Every time you sleep in a bed, you go forward by a day. The passage of time affects Peatrice, the Item Check Girl in the Skyloft Bazaar. Each time you talk to Peatrice after you have slept in a bed, she will become more convinced that you are romantically interested in her. One day, she will start calling you "darling".
If she has started calling you "darling", go talk to her father, Peater, on Bamboo Island northeast of Skyloft if you haven't already. (You can play the Clean Cut minigame there, too, and strum the harp near the butterflies to reveal a Gossip Stone.) Then if Peatrice has started calling you "darling" whenever you talk to her, sleep in a bed on Skyloft until night, then visit her house at night and talk to her father. Her house is on the lower level from the Bazaar, on the southeast side, a red house with a green diamond over the door. If Peatrice's father doesn't say anything about her admirer, sleep until night again and talk to him again. Then sleep in a bed until day, talk to Peatrice at the Item Check during the day, and she asks you to talk to her at home at night.
Sleep until night and talk to her in her house. You can either accept or reject her confession of love. You won't be able to change your choice later. You will still get the same reward, but the outcome will be different. If you accept, she gives you 5 Gratitude Crystals. If you reject, you can talk to her father the next day to get 5 Gratitude Crystals.
Find the Propeller
If you dowsing for the propeller while on Skyloft or on another sky island, you will see that the propeller is somewhere around the Eldin Volcano. You can also learn this from the fortune teller in the bazaar.
The propeller is close to the entrance to the Earth Temple, so when you dive down to Eldin Volcano, choose the Temple Entrance Bird Statue. Go southwest from the temple entrance and look down the slope. Use dowsing to search for the propeller. You will see it near the fallen wooden tower at the bottom of this slope. Go down the slope and examine the propeller to have Scrapper take it to the sky.
Go back to the sky. Scrapper will be flying behind you with the propeller. Go to Skyloft. At the broken windmill on the southeast side of the island, Kukiel's father will automatically fix the propeller for you.
Turn the Windmills to the Light Tower
Use the Gust Bellows on the propeller to rotate the windmill. Line up the notch on the upper part of the windmill with the notch on the lower part. The windmill may keep rotating for a moment after you stop blowing air at it, so stop before the notches line up, then if it isn't lined up, blow a brief gust of air at it to make it go the rest of the way. A blue light will light up on the back of the windmill when it's pointing the right way.
If you didn't rotate the other windmill already, go to the windmill on the southwest side of the island and point it at the Light Tower as well.
A platform will rise on the Light Tower. Climb up to the top of it. A sign there says to play the Goddess's Song. Stand in the circle and strum the harp. Strum in time with the pulsing circle. The tower will shine a beam of light at the area of clouds in the northwest.