Grim Tuesday

Grim Tuesday is the second novel in the "The Keys to the Kingdom" series by Garth Nix. It was published in 2004, in the US first by Scholastic Press and in Great Britain by HarperCollins Children's Books. It is about the adventures of a boy named Arthur who tries to gain control over a magical world called the 'House'.
Plot.
Background.
Arthur Penhaligon is a young boy who has gotten involved with the 'House', a magical world. This world comprises seven parts, each containing a 'Key' (powerful magical objects) and a part of the 'Will' (a being that holds the wish of the absent 'Architect'), under control of a villainous 'Trustee'. Arthur is on a quest to defeat the 'Trustees' and fulfill the 'Will'.
In the first book, Arthur defeated the first Trustee and freed the first part of the Will, which took the form of a lady called Dame Primus.
This book.
Arthur is at home when Dame Primus contacts him. She tells him that in the six months of House time that have passed since he left, the second Trustee, Grim Tuesday, has taken control of Arthur's part of the House that he captured in the first book. Arthur travels to Tuesday's part, called the Far Reaches, with some difficulty, but when he gets there he is mistaken for an indentured worker and forced to work. His ally Suzy finds him and takes him with her towards Tuesday's tower, which holds the second Will fragment and Key.
When they reach the tower, they find that it is inside a giant glass pyramid. A creature called Soot, who claims to have been Tuesday's eyebrow, gives them a diamond to cut through the glass, in exchange for helping it into the treasure tower. Inside the tower, they meet a man called Tom Shelvocke or 'the Mariner', a son of the two divine beings the 'Architect' and the 'Old One', who is working for Tuesday due to blackmail. The Mariner transports them to a world inside a bottle, where they retrieve the second part of the Will.
Grim Tuesday arrives and chases them, but then one of Tuesday's servants tells them that the east buttress of the Far Reaches is giving out, and that it will fall if not fixed soon, leading to the destruction of them all. Tuesday, who has been deposed by the Will fragment, demands the Key to solve this problem; the Will fragment, however, declares a contest between Arthur and Tuesday of creating something original with the second Key, of which the Mariner is judge. Tuesday makes a beautiful tree of precious metals; Arthur, thinking he cannot compete on physical beauty, creates a xylophone and plays a tune he composed. The Mariner declares Arthur the winner, because Tuesday's tree is a copy of an existing sculpture, while Arthur has made something original.
Arthur goes to the eastern buttress, where he first defeats a strong House creature that attacks him. He manages to mend the wall and keep the buttress from collapsing. The second part of the Will merges into Dame Primus, who at his request reverses the effect of the first Key on him so that he stays human. Due to this, he is in very ill health when he returns, and is sent to the hospital. When he wakes up, he finds an invitation from the third Trustee, Drowned Wednesday, under his pillow.