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Cigarette lighter fuse is in the trunk, left side, fuse #20 , yellow one.
Chevy Malibu Classic The fuse for the cigarette lighter and outlet is in the black box under hood in front of driver, location #34 replaced with #20 yellow fuse
Rather than re-invent see http://www.fixya.com/cars/t11556946-power_outlet_not_working
It sounds like you might have drained your battery! When the car is running does the Cigarette lighter produce power? It should be working while the car is running. That would also explain why the car starts sporadically. Try leaving the car on for 40 minutes. That might charge the battery enough.
There is a short in the wire somewhere... Disconnect the battery, get an ohms meter and remove both the wires from the cigarette lighter and check for connectivity. Trace the wires back to where the short is. It is also possible that the socket is bad, look inside the socket, you will see the positive connector in the socket, the ground is the sides... without touching the two with the same probe at the same time, make sure the wires are disconnected and chech with the ohms meter for connectivity between the side of the socket and the positive lug on the back of the socket.
There are actually FIVE cigarette lighter/ Auxilliary Power Outlets shown in the wiring diagrams for your vehicle. The Instrument Panel  Accessory Power Outlet (Cigarette Lighter) is protected by the "LTR" Fuse (# 53 - 20 Amp) in the Power Distribution Center (PDC) in the engine compartment.
The Instrument Panel  Accessory Power Outlet is protected by the "AUX PWR" Fuse (20 Amp) In the Instrument Panel Fuse Block on the left side of the dash.
The Center Seat Accessory Power Outlet and the (two) Center Console Power Outlets are protected by the "AUX PWR 2" Fuse (20 Amp) In the Instrument Panel Fuse Block on the left side of the dash. NOTE: If your vehicle is not equipped with the Center Seat Accessory Power Outlet, or the Custom Floor Console (optional), this fuse position will most likely be empty.
A good thing to check to see if it's just a fuse is try your external mirrors to see if they move, they work off the same fuse as the lighter.
If it is a fuse, and if you are in Australia then the fuse box is next to the drivers right knee area, In America, then next to their left knee. There is also a fusible link built into some cigarette lighters, mine has one. When it blew, the auto electrician was able to resolder the link and get me going again until I could buy a new lighter.
Sounds like maybey someone has replaced the lighter before, and has somehow tied the ground wire into the headlight switch. Short of going through the wiring harness between the lighter and switch, I would just say run a new ground wire and get a fuse tap for the power. This will completely isolate the lighter from the rest of the electrical system. Let me know if this helps :)