How cool is it in the house? If it is about 70 and maybe a degree or two below and your snake has its uth I think you will be alright. I may through a blanket or towel over the unit to insulate it a bit more.
Those lamps are fairly simple. Unplug it and then unscrew the ceramic base from the internal part where you screw in the bulb. You will see the two wires that should be attached with a screw to the terminals. If these two wires are connected well the issue is probably inside the switch. Depending on the unit it may split apart so you can check it. If not and its an emergency you can cut the switch out of the circuit and then wire it back together. THIS CAN BE VERY DANGEROUS AND CREATE A SAFETY HAZARD!!! I would only attempt this if its an emergency and you have some experience wiring such things. It is super important to insulate the two wires and not allow them to arc and create a fire hazard.
If you have a desk lamp with a extending neck it could easily be used until you can get another lamp.
You could also go to a 24 hour Walmart and get a new fixture there. They should have a utility style light that matches what you have in the hardware and paint section. If not you could get a cheap desk lamp with a craning neck.
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