Try this:

1. Unplug the charger cord from the laptop.
2. Remove or disconnect the battery.
3. Press and hold the "Power" button for 30 seconds.
4. Plug in the charger cord.
5. Press power button. Does it come on?
6. If yes then install the battery.

Has that fixed it? Let us know.

Here is link to the User Manual:

Toshiba Satellite L650 User Manual

Dynabook is the new owner of the Toshiba brand

Here is a link to the Maintenance Manual:

Toshiba Satellite L650 Manintenance Manual

If you have a multimeter check that the charger is outputting 19VDC. If not get a new charger.

How old is the battery? They generally last about 5 years if well looked after. It could be a defective battery.