Ms Lann, from Pollokshields, was last seen by her daughter around 9pm in Battlefield Road, Cathcart, near to the Victoria Infirmary, on Monday, April 2.
Her family reported the 47-year-old missing after she failed to contact them, which police said was "not normal".
As reported in later editions of last night's Evening Times, a group of teenagers discovered her body at the premises in Cowlairs Road, at around 2.30pm on Wednesday.
Police today revealed that, following a post mortem examination, there are no suspicious circumstances into Ms Lann's death.
But detectives are working to establish how she ended up at the disused site, which includes a car park.
The building was formerly J Harley's Rider Training Centre, but it closed about two years ago. The area around it is now overgrown and entrances are blocked.
Detective Inspector Brian Dodd, from Baird Street Police Station, said: "The results of the post mortem reveal there are no obvious suspicious circumstances surrounding her death.
"However a full investigation is ongoing to clarify the circumstances which led to her being at a disused building in Springburn.
"It is more than five weeks since she was seen and our inquiries are focusing on what led her to the factory, when did she go there, how did she get into the premises?
"We need answers to these questions and would appeal to the public for help. I would again like to speak to anyone who saw Diane since she was last seen on Monday April 2."
Anyone with information is asked to call 0141 532 4116.