Battery connector doesn't work

There is no standalone connector on the iPad air unfortunately, the connector is part of a cable directly soldered to the logic board. Unless you can find someone in your area who can do microsoldering, you should replace the board which would probably cost you as much as a second hand tablet. You can try blowing it with a compressed air can and clean it with alcohol and a tiny piece of cloth and tweezers in case it just about some debris and dirt stuck inside.