Miss Aim's father, Brian, said he was "greatly comforted" that his daughter was back in Scotland.
"This means so much to me," he added. "We weren't sure exactly when she would be back, and thought we might have to postpone the funeral.
"But it seems much more realistic now that it will happen on Tuesday."
Miss Aim's friend, Alan Robertson, who is originally from Orkney but now lives in Taupo, accompanied her body on the flight.
This was made possible after the Taupo community raised funds to enable him to travel with her.
A murder inquiry was launched into Miss Aim's death after she was found fatally injured in the street.
A memorial service for Karen will be held in Taupo where she died at 11am on Tuesday, to coincide with her funeral.
A memorial fund has also been set up in her name.
Last week, around 100 members of Miss Aim's local community in the Orkney parish of Holm paid tribute to the backpacker at a special church service.
It was attended by Mr Aim, who was accompanied by his mother, sister and brother-in-law.
His wife Peggy and son Alan chose not to go.
Mr Aim had previously said the full reality of what happened to his "bubbly, bright and cheerful" daughter would not sink in until she returned home.
The officer leading the investigation into the killing Detective Sergeant Greg Turner has repeated an appeal for information from the
public.
The probe has been focused on a group who vandalised a nearby college.
Six windows were smashed at the Taupo Nui-a-Tia College at around the time
Miss Aim was assaulted and the vandals have yet to come forward.
- SCOTTISH PRESS ASSOCIATION