I have tried with your suggestion, but unfortunately to no avail. Yesterday around 23:55 I held the shift button while clicking on the "shutdown" command. I have just turned the Surface on, and launched the battery report. The battery loss is around 4% over almost 24 hours:
I will try with the other suggestions, namely the file integrity and the fix proposed by inboxerror2010. At this point, I have nothing (more) to lose.