First of all, you need to understand that "with all his might" ("might" here is a noun, not a verb) is an adverbial phrase meaning "with all his power", so "with" is not connected with the preceding verb (i.e. "demand"). So the last part of the sentence should be read: "...to demand || with all his might || that which he desires". In other words, "demand" is the verb, "that which he desires" is the object (demand what?), and "with all his might" shows the way of demanding (how?)
"that which" means "the thing(s) which". It could also be expressed as "what" ("what" in this case being a relative pronoun, not a question word), but then it would be somewhat less emphatic.
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