Reflowing a CPU is maybe an extreme first option to try.
Does it do this with the charger connected and switched on you didn’t say?
Try a power refresh and check if this resolves the problem. Sometimes the BIOS can become corrupted, preventing a normal startup and performing a power refresh, restores the BIOS to its factory default settings
i). Disconnect the charger if connected and then remove the main battery from the laptop. Here’s the Ifixit Toshiba Satellite A665-S5170 Battery Replacement that shows how to do this.
ii). Press and hold the laptop’s Power On/Off button for a full 30 seconds and then release it.
iii). Reinsert the battery into the laptop, reconnect and switch on the charger and then try turning on the laptop and check.
If still no good then try the same procedure as above but this time remove the RTC battery (supplier example only to show what it looks like) after removing the main battery and dis-assembling the laptop to a point where you can disconnect the battery from the motherboard.
Here’s a disassembly guide that may help.
After it is removed, use a voltmeter to measure the battery voltage. If it is <2.5V DC replace it.
Then reconnect the RTC (Real Time Clock aka CMOS) battery, re-assemble the laptop, reinsert the main battery and connect the charger and try to turn it on.