Once you have defeated all three planets (jungle, ice, and fire), and have all three space knights with you, travel to the asteroid belt in the lower left side of the solar system. You will find the Binary Bard's hidden castle on the asteroid at X-11 Y-84.
if you mean the planet in the asteroid belt then it is X:11 Y:82
When you have all three knights with you, they suggest you look for a giant asteroid, which happens to be at X-11 Y-82 in the Asteroid Belt.
She isn't on a planet.she is on a asteroid (if you are talking about astro-knights island)
No, because you need the three knights to go with you. (see related question)
Once you have beaten all three planets and have all three knights with you, travel to the secret asteroid in the Asteroid Belt (down and left in the solar system). The asteroid is at X-11 Y-84.
On a giant asteroid in the asteroid belt.
You do not get or use a bucket on Astro Knights Island.
When you have all three knights but can't find the Princess, the knights suggest you look for a fabled "giant asteroid". There is one in the Asteroid Belt near X-11, Y-82.
If you have all three Knights with you, proceed to the Secret Asteroid (X-11 Y-82) and rescue the Princess. If not, there's nothing to do but shoot space rocks.
There is no treasure chest on Astro Knights Island.
When you have beaten all three planets and have all 3 knights with you, travel to the giant asteroid at X-11 Y-84 in the asteroid belt. There you can pull out the Golden Key and enter the realm of the Binary Bard.
(See the related questions for each of the three planets, and for the final battle on Mordred's asteroid.)