As suggested the adapter being broken would be the first thing I would test.
Borrow or get a multimeter, and set it to 20v DC (that is the straight line, not the curvy one)
Get a piece of wire (preferably solid and not stranded) and put it in the center of the round plug from the power adapter.
With the adapter plugged in, touch the red lead on multimeter to that wire, and the black lead to the outside metal of the power adapter plug (do not touch either the wire or the outside of the plug with your body, you wont get shocked but it will cause the readings to be wrong).
Compare the voltage to the voltage listed as Output on the power adapter (most are 16 or 19v). If it is much lower then that (or zero) then you have a dead power adapter.
If the power adapter tests good then the likely thing is the battery.
Take the battery out, plug the power adapter in, and see if it will turn on. If it does then replace the battery.
If both adapter and battery are fine then you either have a bad DC jack connection or a bad motherboard.
I had an ASUS laptop that I had to resolder the DC Jack because it cracked.
If the power adapter is bad, do not buy a generic "Replacement brand" power supply. Buy either the ASUS branded one or a DELTA branded one (Delta makes the power supplies for ASUS so these are exactly the same). I have seen far too many laptops fried due to cheapo power supplies). You can almost always find the actual ASUS or Delta ones on Amazon for only $5-10 more then the cheapo ones.