The high-profile case caused shock in the east African country and Agnes Wanjiru's family have long urged the UK government to open a "proper investigation" into her death.
The 21-year-old was found in a septic tank after she reportedly went partying with British soldiers at a hotel in central Nanyuki town, where Britain has a permanent army garrison.
After the first meeting of a British government minister with the family, Defence Secretary John Healey said in a statement that it was "deeply humbling".
Healey will meet Kenyan President William Ruto later on Monday and said he would "emphasise the need to accelerate progress in this case".
The case has been in limbo for years. Kenyan investigations have struggled while the former UK government stonewalled media allegations the army knew about the incident and did nothing.
Healey -- who claimed while in opposition that the previous government was "sitting on their hands" -- reiterated his "determination to see a resolution to the still unresolved case".
He said Britain would continue to offer "full support to the Kenyan investigatory authorities", adding: "Our government will continue to do everything we can to help the family secure the justice they deserve."
- Hearings postponed -
Following the Labour Party's victory over the Conservatives in Britain's elections last year, Wanjiru's family said it was pressing for progress on the case.
Her family said on Monday that while they were grateful for the meeting, attended by elder sister Rose Wanyua and other relatives, they had "waited for too many years and been offered too many empty promises".
They urged the Kenyan and UK governments to act to "bring closure to this matter" and ensure "what happened to Agnes never happens again".
In October 2021, The Sunday Times reported that a soldier had confessed to his comrades to killing Wanjiru and showed them her body.
The report alleged that the murder was taken to military superiors, but no further action followed.
A Kenyan investigation was opened in 2019 but no results have been disclosed, and public hearings due in 2024 were postponed.
At the time of the Sunday Times report, a Labour spokesman told the newspaper the party would "of course" begin an inquiry if it took power.
Since Kenya gained independence in 1963, Britain has kept a permanent army base near Nanyuki around 200 kilometres (125 miles) north of the capital Nairobi.
The British Army Training Unit in Kenya (BATUK) is an economic lifeline for many in Nanyuki but has proved a lightning rod for criticism.
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