Hi,
In case some might me interested, I just got accepted to the MSc in Finance programme at LSE.
My profile is as follows:
Nationality: Norwegian
GMAT: 720, 6 AWA
GPA: 3,688 (A Norwegian GPA of 3, or "B", is by LSE considered equal to an american GPA of 3,5)
WE: None what so ever.
References: At most average; we neither rank student nor have a close relationship with our professors.
The average GMAT score of the MSc in Finance students last year was around 690, so 680 should not be THAT big of a long shot.
I do believe the references are are very important. I have heard (from former alumni) that it is important for LSE to know that you are a top student from a top institution in your home country.
Also, your grades are VERY important. You should check LSE's website in regards of how they rate your country's grade scale.
I am curious about where you are from, as you are applying for the same programmes as I did. Are you scandinavian?

Good luck with your applications.
Cheers