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The children in the orphanage are all available for adoption. It's the second option in the dialogue box, though you can also select first the "Tell me about yourself" option to hear the child's story. The child will express excitement, after which you can head back to the home you chose earlier when speaking to Constance Michel. The child will be there when you arrive. You can adopt up to two children. They will always be there when you return. You won't have to feed them or worry about them being killed by NPCs, since your children can't die. Children whose parents are killed by the player or by other NPCs will eventually appear in the Honorhall Orphanage. Adoptable children are as follows:
Find a child whom you wish to adopt and speak to them. Select the "I can adopt you, if you want." option. Select the "Yes I'm sure...daughter/son." Interact with your children by visiting your house. Consider adopting a child from a dead NPC.