Dohna has been tightening security and challenges you to steal the contents of the treasury.
Where to Start the Quest
Talk to Dohna when she is standing in the eastern part of Gerudo Town to start the quest.
How to Complete the Quest
Go to the second floor of the Gerudo Palace, then go up the stairs behind Seera's throne. Go west and you find the treasury building at the end of the roof. Use echoes to get on top of the building, then use a Holmill echo to dig into the darker patch in the roof. Then fall through the hole that the Holmill dug.
The chest to the north contains 10 Chilly Cactus. It's not the treasure that you are supposed to steal. Be careful going to that chest. If you make a loud noise by falling to the ground, the guards will come check what the sound was, and you'll be sent to the entrance of the building.
Go east from where you first arrived in the building, and don't let the guards see you. You can safely go up the left side of the stairs without alerting the guard that is looking at the statue.
To the north, there is a soldier walking back and forth. Watch her pattern before you try to go north.
The final soldier just looks left and right. You will have to create an echo, like a Pot echo, and throw it left or right (but not too far to the south), then when the soldier is going toward it to look, sneak behind her and go north and open the chest. There will be a cutscene.
Quest Reward
You get to keep the Silk Pajamas that you stole from the treasury.