Dom, hopefully this will help.
On Scotland's People, if you search for a death for Jessie Fraser, other name McKenzie, in 1916 you will be able to see her death certificate. Reference is 685/03 0221, and the registration district is Edinburgh City/Midlothian.
You may need to post that certificate on the deciphering board as well, because I could swear her mother's maiden name is given as Dooling! It is your Jessie though, she is widowed and it gives her husband's details as well as her parents