I have a Satellite A660 and my battery doesn't want to charge...it sticks to 0% and when I unplug the cord, the computer shuts down. After many computers restarting and failing finally understood what the problem was. The problem was my BIOS which is Version 1.20.
When I try to install the BIOS, I always get these three errors:
BIOS Version Above 2.00
Battery Capacity more than 20%
AC is attached
Not sure if this would be the same case on Sat A660 but you should definitely try this out. You can try to re-insert the AC-Adapter for several times until the battery is detected and the unit powers on. If this does not help please remove the Main Battery and try again.
Then update the BIOS
It seems your model is built for Canada. I could find it on the Canadian Toshiba site.
No information about revisions information of the BIOS in your model was provided though.
However, as it seems many hardware platforms are similar, however NOT always compatible, I looked at the US web which more generously provides revision information for the BIOSes and I found the BIOS which probably is similar or related to your BIOS, and there are in fact some battery-related fixes.