He initially wrote back to her saying her works were "splendid".
But a few weeks later, Tolkien's "life and affairs were thrown into chaos" when he was hospitalised, Dr Tankard said.
Miss Fairburn later explained to Tolkien she had been "sleeping on the floor of a condemned basement", and faced significant hardship and debts.
But her hopes of being able to illustrate The Lord of the Rings were to be dashed.
She later emigrated to Australia, where she has lived for more than 40 years, and the academic world knew nothing about Tolkien's praise for her or about the pictures she had produced.
All that quickly started to change after Dr Tankard travelled to Castlemaine, a small city in Victoria, Australia, about 120km north of Melbourne, in late 2010 to visit family members.
Told an artist living there had apparently had links with Tolkien, Dr Tankard went to see her, and was shown letters from Tolkien, and knew he was "on to something".