I doubt this is relevant anymore but I thought I'd share just in case someone else runs across this:
I opened up my Spin 5 to clean the fan and had a similar problem, though my battery LED only flashed briefly when I connected the power. I had unscrewed the battery and lifted it up, but didn't touch the connection.
To recover, I:
-Cleanly shutdown Windows and wait until the computer is fully off to disconnect power. Not sure if it mattered, but I figured it couldn't hurt.
-Disconnect and reconnect the battery.
-Replace all the case screws. I suspect that they electrically connect the shielding on the bottom plate to the laptop, and that this was the actual problem in my case at least.
After that, it booted up just fine on battery power.