Assuming that you used the bucket of water with the drain, the key should be here in the mud
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The key is stuck down a drain behind the kitchen in the castle. Go get a bucket or jug and fill it with water at any water source and the sink in the kitchen gives you water (if you did not bring a bucket, up the stairs there is a rooms at the corner and you can get a bucket in there). Use the bucket of water on the drain and the key will be washed into the sewers. Enter the Varrock sewers using the manhole located immediately east of the castle's keep. Follow the north-western path until you get to a stream and go to the small corner. Search for a rusty key on the ground; it will not appear on the minimap. Grab it.
Sir Vyvin can be found on the second floor of Falador castle, to the east.
IF SIR LEYE IS NOT THE ONE YOU HAVE TO KILL FOR THE BRAINS TEST, THIS ANSWER IS WRONG To beat sir leye, you must be a female.
He is easy to kill only level 20, you have to be a woman in Runescape to get to him though, my guesses your doing the recruitment Drive Quest.
You can find this guy in the West side of Varrock palace, ground floor