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

Sir Simon is finally able to get the eternal rest that he so desperately desires only because Virginia Otis agrees to help him by praying on his behalf. Virginia Otis is the only member of the Otis family that is not antagonistic toward Sir Simon. Because Virginia is kind to him, Sir Simon explains his situation to her. He tells her that he desperately wants to get his eternal rest, and that he has been searching for it for 300 years. He tells her about the Garden of Death, and Sir Simon explains that Virginia is a person that is capable of helping him cross over to his eternal resting place.

"Yes, death. Death must be so beautiful. To lie in the soft brown earth, with the grasses waving above one's head, and listen to silence. To have no yesterday, and no to-morrow. To forget time, to forget life, to be at peace. You can help me. You can open for me the portals of death's house, for love is always with you, and love is stronger than death is."

She is pure, innocent, and beautiful, and that is exactly the kind of person that he needs to intercede on his behalf.

"They mean," he said, sadly, "that you must weep with me for my sins, because I have no tears, and pray with me for my soul, because I have no faith, and then, if you have always been sweet, and good, and gentle, the angel of death will have mercy on me. You will see fearful shapes in darkness, and wicked voices will whisper in your ear, but they will not harm you, for against the purity of a little child the powers of Hell cannot prevail."

Fortunately for Sir Simon, Virginia Otis agrees to pray on his behalf.

"I am not afraid," she said firmly, "and I will ask the angel to have mercy on you."

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.