Hi @gye7293 ,
Does the laptop indicate that the battery is charging when you just connect the charger to the laptop?
If not, as it doesn't start with the charger connected, then you may have a faulty DC-In jack.
It may be faulty or it may have become disconnected from the motherboard.
Here is a video that shows how to access the DC-In jack.
Here is a link to just one supplier of the replacement part.
It is not a recommendation to use them. It is shown to give an idea of the cost of the part. There are others online that may suit you better. Just search for Asus Q400A DC-In jack part to get results.
You will need soldering expertise and tools to effect the removal (if necessary) and replacement of the part.
If this seems too daunting, contact a reputable, professional laptop repair service and ask for a quote to repair the laptop. You already know the cost of the part. ;-)