It seems like the most likely explanation would be that something is wrong with the new battery you installed. Given that the computer boots, even though it doesn't stay on long enough to get past the login screen, that's a good sign that you didn't break anything during the repair process. I would recommend if you have a multimeter, try probing the contact points of the battery and see if it registers any voltage (somewhere around 11.25 would be normal). If not, I would return the battery and describe your issue and see if iFixit can handle sending a replacement or give specific advice about their battery. It may also be that one of the cells that make up the battery has come disconnected, or there is an issue with the small bms board that is part of the battery.