The thing was... The hardware switch was on the "on" position.
This is how to fix this problem.
1. Open up a terminal and run the rfkill command
We'll be listing the devices blocked by rfkill. Devices that are soft blocked are being blocked by the software. Hard blocked devices are devices blocked because of a physical switch. This is usually the case, except here, Ubuntu is erroneously reporting the wireless network interface being turned off because of the hardware switch.

Run the command :
rfkill list
2. Unblock all the devices being blocked
Each item listed starts off with a number. For example, the first item listed for me is the "sony-wifi: Wireless LAN". We can tell this is device 0 because to the left of the name, is the number 0.
The numbering system for listing the devices are necessarily numbered consecutively, so make sure you grab the correct device number for the following command.
Run the command :
rfkill unblock # (where you replace the # with the number of the device we're unblocking)
Repeat the same command for all the blocked wireless devices.

3. Check your work
Run the rfkill list command again to check if your wireless device has been unblocked.
With your Wireless LAN now unblocked, the "Enable Wireless" option should now be un-grayed. If the option is still grayed out, refresh the options by disabling networking (un-check Enable Networking) and re-enabling networking (re-check Enable Networking).
After networking is enabled, you should be able to select "Enable Wireless".
Video tutorial: http://youtu.be/KjWggBx23rY

