She used ashes to make a help sign in the snow, and fashioned snow shoes with sticks during the ordeal, Fairfax reported.
She had attempted to get out of the area, but was unable to, due to her mental and physical condition, Czech Republic honorary consul Vladka Kennett said.
"Nobody can prepare you for this," the woman told police.
Otago Lakes central area commander Inspector Olaf Jensen said the woman made the right decision to stay at the hut.
"Some of the comments asking why she wasn't found are unhelpful. No one had been through the area and because of her physical capability, she wasn't able to walk out.
"Given her experience, and the avalanche risk, she made the decision to stay in the hut, and that was the right decision."
Jensen said the woman appeared to have suffered from minor frost bite. She was examined at hospital after being found yesterday but didn't require treatment.
The woman is now staying with Kennett in Glenorchy.
Kennett told NZME she was recovering very well, considering what she went through.
"It's been very emotional, as you can imagine, but she's handling the situation reasonably well."
She said the woman would be going home as soon as she could.
Fairfax reported that the woman and man became disoriented on the second day of their walk and slipped down a stone cliff. The man fell further than the woman, who was able to get to him, but could not free him from branches and rocks in which he was tangled.
Kennett told Fairfax the woman heard his last breath before he died.
After the fall, she spent three nights in the wilderness before making it to the hut.
She finally reached it on the fourth day, breaking in through a window.
The Lake Mackenzie hut is self-contained with four bunks, and cooking and heating facilities.
There was a radio in the hut, but she was unable to make it work, Jensen told the Herald.
Jensen described the woman's ordeal as "traumatic".
Her and her male partner, who were both in their late 20s or early 30s, had some tramping experience and were reasonably well-equipped but didn't seem to have told many people they were doing the tramp, Jensen said.
Relatives in Europe raised the alarm that the couple was missing on Facebook and it was the Czech Consulate which raised concerns with the police yesterday about the pair.
Photos and the couple's car registration are understood to have helped authorities track the pair to the Routeburn Track.
Police first found the couple's vehicle at the Routeburn Track carpark and then initiated a search and rescue operation, which included the use of a helicopter.
Rescuers searched huts and areas of high risk in avalanches, Jensen said.
Police believed the man fell about 2km from the hut where the woman was staying. The woman found him after he fell and spent a number of nights in the open after his death before arriving at the hut.
She didn't go back to him after he died, Jensen said.
Search and rescue was now involved in a recovery operation for the man.
Police postponed the search tonight.
"Due to weather conditions and the nature of the terrain, it has unfortunately not been possible today to recover the body of a Czech man who died following a fall on the Routeburn Track," police said.
"Conditions permitting, the recovery team will make another attempt in the morning."
The man's next of kin have been advised of his death.
The pair weren't carrying a locator beacon or a tent with them on the track. They had been in New Zealand since January for travel and a working holiday.
Department of Conservation Wakatipu operations manager Geoff Owen was unsure if the hut would have had food in it, as the department discouraged wardens from leaving food during the winter months because of a rodent issue.
Owen said the woman broke in to the warden's hut, "and good on her for making that decision."
Ultimate Hikes had a lodge nearby the hut which Owen believed the woman may have accessed for food as well.
Owen said the pair might have been found a lot sooner, and there could have been a different outcome, if they had been carrying a locator beacon.
However, he commended the woman for her actions considering the circumstances.
'Five weeks - it's just unbelievable'
The Lake Mackenzie hut is about 43km away from the Glenorchy end of the Routeburn track and takes about 5 hours to reach from the Divide on Milford Rd.
This morning Otago Tramping and Mountaineering Club spokesman Ian Sime said it was unlikely the couple had left tramping intentions or signed in before starting their hike.
He said it normally just three days to complete the popular walking track.
"She can't have filed an intention form at the start because if she did someone would have been looking for her within a week."
"Five weeks - it's just unbelievable," said Sime.
Given she had stayed in a warden's hut she would have been warm and had access a large supply of food.
"If she was in the warden's hut she would have been okay," he said.
The Lake Mackenzie warden's hut was locked at this time of year so the woman must have broken in, Sime said.
There were no conservation rangers stationed along the track outside the Great Walks season.
The Great Walks season ended on the 27th of April this year and would restart on October 25 according to DoC's website.
Sime said at this time of year it was possible snow conditions would make the track difficult for people to pass through.
Flushing toilets at the site were shut down for the off season, replaced with pit toilets and gas was not provided, with DoC recommending online that trampers bring their own cooking stoves.
Running water was also turned off inside the huts but could be accessed from the outside water tank or by melting snow.
Ultimate Hikes general manager Noel Saxon, who runs guided tramps of the track, was surprised it took a month before the woman was found.
He said even during the quieter winter season he would have expected another tramper to have come across her earlier.
Heliworks Queenstown Helicopters general manager Richard Mills said his crew flew the woman out of the area.
He said the woman was "ecstatic" when she was rescued.
There had been no instructions about recovering the man's body.
He said the tragedy could have been averted if the couple were carrying a personal locator beacon.
"If any good can come of this it is the message to carry PLBs. They can take the 'S' our of SAR [Search and Rescue]."
The Routeburn Track stretches for 32 kilometres linking the Mount Aspiring and Fiordland National Parks.
It can be entered at either end with one entrance at Glenorchy and the other at the Te Anau-Milford Rd.
It is ranked as an intermediate track by the Department of Conservation.
Ultimate Hikes general manager Noel Saxon said at this time of the year there's likely to be a lot of snow, markers would be covered and walking out would be very difficult.
Mr Saxon said it's remarkable the woman survived a month, and that it would've been a very lonely experience.
