Hello all!
I think I GOT SOLUTION for this problem.
I have the same issue in my VAIO SVT13115FDS. The battery was show 100% charged but when I unpluged the power cord the laptop just turn off.
Actually the solution is very simple:
1 - Turn off the laptop;
2 - Disconect all cable from it;
3 - Close the lid;
4 - Put the laptop upside down;
5 - Sarch for a small hole (the reset button) on the back of the laptop;
6 - Grab a paper clip and introduce it in this small hole;
7 - Press just enough to feel push the button for 10 seconds.
Now try turn on you laptop...
Enjoy!