It looks like you have a MacBook Air Mid-2012 mode, but that is just an educated guessl. You'll want to confirm exactly which model you have for sure though by running your serial number through here:
http://checkcoverage.apple.com
Most MacBook Air's do have a replaceable DC-in board (aka Magsafe board) that is easy to replace, but it's possible what is happening is that your bottom lid is actually shorting out your DC-in board.
I would take off the bottom cover and pound down that corner of the bottom lid that appears to be digging into the DC-in board and seeing if just flattening it cures the problem (if it's a short-circuit), otherwise that tiny (inexpensive) board needs to be replaced

For example, to pick up a replacement for a 2012 model you could go here for $21:
https://www.powerbookmedic.com/MacBook-Air-133-Magsafe-Board-p-24200-o-3421.html