Chosen Solution
Chronological list of fails: First: During normal use suddenly reboot Than: Boot loop Than: Adapter showed no more light Than: Macbook is dead Than: New battery - nothing changed (1406) Than: New I/O board - starts for two seconds when connected to adapter (which shows no light), than goes dead. Also no battery charging. Any ideas?
That sounds pretty bad, you’ve attempted all reasonable steps in a DIY approach, there’s really nothing left for you to do except replacing the logic board in my opinion. Unless you have an i7 with 8GB soldered RAM I’d be very hesitant even to recommend a board repair as when Macbooks Air go bad, they often suffer fatal issues involving the i5 chip itself or related circuitry.