Archimedes' Lever is the sword wielded by the original Storm King. Forged by master-Spark Van Rijn, along with being powerfully electrified in some fashion, it was "bound to its owner's soul", allowing him to sense and activate it from (at least) a fair distance away. According to Violetta, the blade was also sharp enough to "cut the day out of the night", whatever that means.
The Jägermonsters refer to it as Slasher.
The Storm King was somehow physically separated from the weapon before disappearing from the historical scene; when first seen in Agatha's era, it is in the possession of Storm-King descendant Tweedle in Paris. When it is reactivated, Tweedle Sparkily attempts to break its connection with the freed/revived Andronicus and then use it against its former(ish) owner. It works to the extent that Tweedle is able to cut part way into the Storm King's remaining weapon. (See below.)
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The sword's name is a reference to a quote from the ancient Greek philosopher-scientist-Spark Archimedes: "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world."