How did Sir Simon get peace and become able to sleep?

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Sir Simon the ghost is able to make a connection with Virginia when she stumbles across him one day in the tapestry room. Because she is kind to him, he is able to explain to her how desperately he would like to go to his final rest—to finally be able to die, rather than to continue on forever as a ghost. However, he has sinned in murdering his wife. He needs someone pure to intercede and pray for him. Virginia, who is pure, agrees to do so.

Although it is hard on her, Virginia's prayers on Sir Simon's behalf are successful, and he is able, finally, to die. He is given a big funeral, attended by the Otises and the Cantervilles. 

When Virginia sees the withered almond tree in bloom, which a prophecy says will happen when Canterville Hall is at peace, she knows that Sir Simon is finally forgiven and at rest.