I have an 11" MacBook Air 6,1 from mid 2013 that I found discarded and had restored but now seems dead again.
I got the machine with a broken screen backlight and missing SSD. I took it to the Genius Bar and they diagnosed only those two problems. I attached it to an external monitor and USB 3.0 hard drive and installed El Capitan and used it for a couple of days. Everything worked except the webcam. Though the machine had sat for several weeks with no power before I acquired it the battery life turned out to be very good.
I put it to sleep when the battery charge indicator had just gone red and did not have access to a charger/power supply for over a day.
When I was able to borrow a friend's power supply the light did not come on. After finding another Mac user we plugged each other's Magsafe into each other's machines and both lights came on. When I went back to the first charger on my machine the light then stayed amber but the machine would not boot and would not even make the startup "chime" sound. I kept it plugged in all day yesterday and the charger light never went green.
I tried resetting the EMC to no avail. I can't seem to find information on this specific problem online. I can't get a reservation at the Genius Bar for another five days.
Best Answer
I had a similar thing happen to my old Macbook. I discharged the battery all the way down and then couldn't recharge it for about two days. When I finally could charge it, it wouldn't charge or turn on. It turned out to be the battery. You can probably get a replacement battery from Apple (if it turns out to be the battery), but I found cheaper options on Amazon.