You can usually unscrew a broken base with a pair of needle-nose pliars. Just get a good grab on the metal and then you should be able to work it out. Unplug it first though, yikes! I don't want someone getting electrocuted because of my advice! If you don't have any tools around, I've heard you can cram the cut end of a raw potato into the broken part and it usually sticks well enough to be able to unscrew it that way.
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