Elsie Frost, 14, was stabbed to death on her way home from her sister's house in Wakefield on October 9, 1965.
An inquest took place between January 4 and 11, 1966 when the coroner recorded a verdict that Elsie had died of multiple stab wounds. The coroner wrongly stated Elsie had been murdered by Ian Bernard Spencer.
Elsie's siblings Colin Frost and Anne Cleave, along with legal representatives, will appear at the Royal Courts of Justice on Tuesday to ask the court to consider their application to have the original inquest into Elsie's death quashed and a fresh inquest ordered.
Mr Frost, 60, said: "Tuesday is a pivotal day. If the judges rule no, then there is no way forward and the decision is out of our hands. If that is the decision there will be no justice for Elsie and her killer will have got away with the perfect murder.
"This could technically be the last chance for us to find out what happened to Elsie.
"All along we have been fighting for justice for Elsie who was brutally and needlessly murdered. Her life was taken away from her family and friends."
Detectives re-investigating Elsie's murder since 2016, were preparing to charge convicted child killer and rapist Peter Pickering after new evidence came to light, but the 80-year-old died in March 2018, after being taken ill in the secure psychiatric accommodation in Berkshire where he was held for more than 45 years.
In the week before his death, Pickering was found guilty of the violent rape of an 18-year-old woman in Sheffield in 1972.
The crime only came to light as a result of the cold case investigation into Elsie’s murder by West Yorkshire Police.
In October, last year Mr Frost and Mrs Cleave kick-started a legal drive for a new inquest into her death so the public can finally learn the truth about what happened and the family can get “closure”.
Elsie, 14, was attacked from behind and stabbed in the back and head as she walked through a railway tunnel off a canal towpath in Wakefield in October 1965.
Mr Frost says his family were "torn apart" by Elsie's death and each individual has had to overcome their own emotional hurdles. He says his parents, who have both since died, lived a lifetime of guilt without knowing what really happened to their daughter.
"We have all had to deal with our own emotions and face different hurdles," Mr Frost said.
"Elsie was my third mum when we were growing up. She taught me to tie my shoelaces when I was five and would take me to children's clubs.
"She was just lovely, she would have been very very special. The harsh thing about this is no one will ever know her true potential.
"I have three grown-up children and four grandchildren - Elsie deserved to see them. She never grew up to be a mum herself or carve a career for herself."
The family say they are also want the result of the first inquest quashed for the sake of Ian Bernard Spencer.
Mr Frost said: "He has had to live with that for the rest of his life and that is just wrong. The actions by the coroner that time were wrong and led to the wrong conviction."
"We want justice for everyone involved."