Does the laptop show that it is charging when the adapter is connected to it?
If not check that there is 19.5V DC at the output cable plug of the adapter. Just because the adapter light is on doesn’t mean that there is voltage at the output and maybe the battery has gone flat because of this.
If so it may be a faulty power button or power button and volume-buttons board or board cable connection.
Here’s the service manual for the laptop, taken from this webpage
Go to p.38 to view the necessary pre-requisite steps and the the procedure to remove/ replace the power and volume-buttons board.
This will allow you to check the cable connection and the button etc.
If it is faulty, replacement boards are available online. Just search for Dell Inspiron 5378 power and volume-buttons board to find suppliers that suit you.
Since there is no indication that it is charging, you need to check if the DC-In power jack in the laptop is OK. Sometimes the centre pin of the jack breaks internally and the power circuit is therefore open circuit and the laptop doesn’t get any power from the charger and also the battery eventually goes flat if it is running on the battery and the laptop won’t start.
Use the service manual I linked and go to p.53 to see the necessary pre-requisite steps and then the procedure to remove the power adapter port from the laptop. You only have to disconnect the port’s cable plug from the motherboard and then use an Ohmmeter to test the connections from the socket to the plug to make sure that they are all OK.
IF the centre pin connection has no continuity from the centre pin to the plug then the adapter port (aka DC-In jack) will have to be replaced.
Here’s an example of what it looks like and how much it may cost. There are other suppliers online. Just search for Dell Inspiron 5378 DC-In jack to find suppliers that suit you. Looking at the image of the DC-In jack, the centre pin should connect to the 3 red wires on the plug and the outer ring connector of the jack should connect to the 3 black wires of the plug