Sorry I only now saw this post but the same thing just happened to my MacBook Pro 13" which has the 'non-removable' battery.
It would not start up and had no charging indication on the power cord LED or on the charge status indicators.
Initially I tried resetting the SMC but that didn't work. So I removed the bottom cover, located the battery connector cable (next to the memory modules) and carefully pried it upwards and off using a mini flat screwdriver. After pausing around 10 seconds, I re-attached the connector and immediately the charge light came on and the system started (fans, tone, but no screen initially). I then held down the power button until the system shut off again and verified the charge indicators on the body as well as charger cord were and they were. I replaced the bottom cover and performed the SMC Reset (with charger plugged in and attached to the laptop, press left side Shift, Control and Option keys whilst simultaneously pressing the Power button for a few seconds then releasing them together). Then I started it normally and it did just that :)
I hope this is helpful as this is basically like resetting the older systems with removable batteries by removing the battery. In this case it is just removing and reconnecting the battery connector.