I never got a letter about adopting children. When I go to the Honorhall Orphanage the girl keeps running around saying "Mercy!" because I killed Grelod the Kind. Can I still adopt children?
In addition to adopting from the orphanage in Riften, you can also adopt some children in other cities as well:
See the Elder Scrolls Wiki for more information.
You need to
To calm down the kids in the orphanage you can use a calm spell or shout.
I believe that the Hearthfire DLC, purchasing the child's bedroom and receiving the letter of adoption are all necessary prior to the adoption dialog options appearing when in conversation with children.
Go and wait a few days for Constance to calm down, otherwise, do the below.
Following Children to Adopt Besides the Orphanage
4.Solitude Katla's Farm: Blaise
You said you killed grelod the Kind in order to adopt children and Constance is still hostile [I don't mean the NPC is attacking, I mean madness]. As I explained first in my answer, to wait a few days for her to calm down. You don't need the letter to adopt from the orphanage, but you do need to wait for her to calm down. if this still occurs, download the Unofficial Hearthfire Patch to fix this.
Anyways don't worry about the orphanages lady Constance, go to the 4 cities and talk to the kids. And if you're new the the DLC, here's a good video for you: