Try disconnecting the battery. Reconnect the power adapter & power on the laptop. Then reconnect the battery while the laptop is still powered on. See if the battery begins to charge. Make sure to wait until the battery charges to at least 10% (more is better), otherwise you risk permanently damaging the battery. Just make sure to keep any metallic & conductive items & surfaces from touching the exposed internal circuitry while the bottom cover is removed during this procedure.
I have gotten some batteries to start charging using this procedure.
With a MBAir, there is also a possibility the I/O Cable connection on the MLB or I/O Board is loose. Also check the I/O Cable connectors & the matching connectors on the Logic Board (MLB) & I/O Board to see if any corrosion is on the inside or on the external solder joints. You may also want to look for signs of corrosion elsewhere on the exposed Logic Board, especially along the back edge of the Logic Board and also the Trackpad connector & cable on the Trackpad. If you see any signs of liquid residue anywhere, then it is possible there could be more liquid residue or even corrosion on the other hidden side of the Logic Board that cannot be seen until the MLB is removed.
You should also try the suggestions in the following Apple article for troubleshooting charging issues:
If your MagSafe cable or power adapter isn't working - Apple Support
Also make sure to unplug the power adapter from the electrical outlet for a minute in order to reset the power adapter's internal circuitry. And try an SMC Reset.
You should be able to get the Magsafe LED to light even with the battery disconnected. If this does not happen, then the battery is irrelevant in the troubleshooting process. This older laptop should power on without the battery connected.