Disconnect everything except the charging board and fan, then plug the charger in. If it is not booting, it will be a logic board related issue; remove the board and check the bottom for liquid damage, especially around the SMC area, which commonly is affected in MacBook Air 13 models.
Try the above and let us know what you find. What is the past history of this too (was it repaired before)? If you have any experience soldering we can then guide you on what to measure and replace on the board.