Hi @surfalia ,
Use a voltmeter to check that there is 19.5V DC on the charger plug. A green light on the charger doesn’t necessarily mean that there is a voltage output getting to the laptop as the charger’s cord/plug could be faulty.
If possible try using another known working compatible charger and check if the laptop works.
If the charger is OK there may be a problem with the DC-In jack - supplier example only.
Use the Voltmeter to check if the voltage from the charger is getting to the motherboard.
Here’s the ifixit Sony Vaio PCG-71312l Repair guide which should help to gain access to the DC-In jack.