Chosen Solution
MagSafe charger still lights up green first and then turns orange when plugged in, but after 24 hours of constant charging, it still stays orange. I’ve done all the simple things (resetting SMC) and others, but I’ve come to my last resort. Do you have a solution?
Disconnect the battery following this guide to Step 6 MacBook Pro 13" Retina Display Late 2013 Battery Replacement. Then with nothing connected hold the power button for a good 15 secs. to fully discharge the logic board. Now carefully plug in your MagSafe charger the system should spontaneously startup, let it finish booting and then shut it down properly. Now disconnect the MagSafe connection and plug the battery again and then the MagSafe charger let it charge up and then see if it restarts, if not you need a new battery.