Millane from Essex, England, was last seen alive in Auckland, on 1 December.
The alleged killer appeared in court earlier this week. A a post mortem examination, meanwhile, is being carried out to ascertain how and when the 22-year-old Millane died.
“Now they have to put forward a prosecution case that will withstand scrutiny to get justice for Grace and her family,” the New Zealand Herald quoted Police Association president Chris Cahill as saying.
“It’s a hell of a lot of work. They are really unusual to deal with when as a detective – on one level they are the ultimate investigation to work on and in some ways the ultimate honour.
“You go and stand up for victims who can no longer stand up for themselves. No matter what you do, you can’t turn back the clock and bring that person back, so you’re starting from a really hard position.
“The best thing you can do is ensure they get justice and to do that you have to do the best possible inquiry you can.
“On these investigations, everyone is so motivated to do the best possible job to ensure the team gets the best possible result.
“You never get people arguing over the long hours they have to work or what roles they are assigned – and some roles are better than others, there’s a lot of hard slog.
“For example, going through CCTV is a hell of a draining process – sitting there working for hours and hours trying to spot the one thing that’s of value to the inquiry.
“But they all know that these little pieces of information are important to getting a result.”
Millane’s body will be released to her family after the post mortem.