Hi @mandabrownell
If you have a DMM (digital multimeter) check that there is a steady 19.5V DC on the plug of the charger's output cable i.e. the plug that is inserted into the laptop.
If you measure the correct voltage try flexing the charger output cable at the body of the charger and also at the plug end in case it is a fractured cable.
If you don't have a DMM try using a compatible charger and connect it to the laptop and check if the power light turns on and stays on.
''If the charger is OK'' it may be a faulty [https://www.parts-people.com/index.php?action=item&id=29914|DC in jack] (supplier example only) in the laptop that is the problem.
What can happen sometimes is that the centre +ve connector in the jack fractures internally and this can cause an intermittent connection to occur between the charger plug and the motherboard.
With the charger plug inserted into the laptop and the charger powered on, try ''gently'' moving the plug up and down and side to side and check if the light goes on and off or may stay on if the plug is held in a certain position. This is only to verify that there may be a problem with the jack.
Here's the [https://dl.dell.com/content/manual47360001-dell-latitude-3400-service-manual.pdf?language=en-us|service manual] for the laptop. Go to p.106 to view the procedure to remove/replace the power port adapter (aka DC-In jack). This will allow you to unplug the jack from the motherboard so that you can test the from the input to the jack to the wires on the cable plug that connects to the motherboard to prove continuity through the plug.
If the plug is faulty it will need to be replaced as it cannot be repaired. Search online for ''Dell TM5N3'' to find suppliers that suit you best.
If the charger and the DC-In jack both test OK then there's a problem on the motherboard and further testing would be required to find out what's wrong.tests OK