Some thoughts ....
I was thinking along the lines of the Millie Marshall whose birth is indexed as Mille Marshall.
That lass was born 1895, so could be a friend or relative of the Simpsons, and may have known Frederick Leslie as a school friend, as they would be of similar age.
Millie's letter seems to suggest there were several people keeping up correspondence with Frederick Leslie P during his hospitalisation in England. The letter was dated 1917.
Re the birth registration for Millie, born 1895 The ref for the birth cert 3807.
Parents recorded for Millie as William Thomas Marshall and Jessie Lang.
There are many other clues in the file for Frederick Leslie Pearson, despite it being in some dis-order.
Cheers, JM