Although that battery appears to be in poor condition, your machine should not be shutting down.
Your machine is not drawing any power from it nor does it appear to be charging.
Let's do a couple of things. Lets see if we can get it to interact better.
Disconnect the battery from the logic board and the mag-safe. Press and hold the power button for 10 seconds. Now reconnect the battery to the logic board. It should spontaneously reboot. See with Coconut battery if it doing anything.
Disconnect the mag-safe and see if it shuts down. If it does not then let the machine run on battery power until it is completely depleted. Then recharge it fully.
These measures failing, replace it with a fair good battery $60+ and we'll see if the machine will talk to the new one.
If this is the same machine you replaced the logic board on, the problem may be with the new board. Please tell us why you replaced that board, I've forgotten.
Also if you have replaced that board you may need to reset the PRAM so all the components know each other.
Press the power button and then press command-option-p-r. You have to make sure you get those keys pressed before the gray screen comes up or it won't work. Hold those keys down until your Mac reboots again and you here the startup chime.