MacBook Air A1237 Charging Problems

I own a MacBook Air A1237 (The "Original" Model) and recently I have really weird problems regarding the battery and charger. The laptop won't power on without the charger plugged in. If the charger is taken of the charging port during laptop's work it abruptly shuts off. When I power it on with the charger plugged in it boots normally. The icon on the menu bar shows "X" and when I click on it shows this: "No batteries available. Power source: battery." Strange, right? When checking coconutBattery in all the battery related spots it shows "N/A". The battery has only 3-5 cycles so I doubt the battery is dead. The battery information isn't useful at all, because it doesn't show any battery information but it says the batteries are installed. I've done the SMC reset, PRAM reset, NVRAM reset, SMC reset without the battery but still nothing. The laptop was held in a room temperature bag for half a month. There is no physical damage except for scratches on the physical body. The MagSafe connector light goes green-orange-green and stays on green forever until a SMC reset or unplug. The battery connector is fully plugged in without any bent pins and the same goes for the charging port and the charger itself. I hope I can receive help from the Apple Forums Community.

- Matthew