The answer is in the question, Varrock Castle (Palace).
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It is in the kitchen. There is a table and a range.
By searching first in the sewer (after using the bucket of water on the drain pipe right outside of Varrock Palace's kitchen), next from Sir Prysin in the Varrock palace and from the Wizard Traiborn in the Wizards Tower.
You climb down the manhole east of Varrock Palace.
In varrock by the palace as you walk round ( the same way that leads to varrock church - east side ) there is a "!" Symbol, that marks the palace sewers.
To the southwest of the Varrock palace, look out on your minimap for a dungeon symbol (a red curved arrow). That points to a "manhole"; enter it.