"The tone with which he spoke drained Rhonin of all animosity." [correct]
The prepositional phrase "with which he spoke" modifies "The tone," and becomes part of the complete subject ("The tone with which he spoke"). A native speaker would never say "The tone which he spoke," because "the tone" refers to how something was said in this instance, how it was spoken. However, if you wanted to eliminate the prepositional phrase, you could correctly write,
"His tone drained Rhonin of all animosity." [also correct]
Does this difference make sense to you? Here are two more examples in response to your comment.
"The force with which he hit him killed his opponent." [correct]
"The finesse with which he wielded his sword was unequaled among men." [correct]