“Surely when the Bible describes our Lord as “King of Kings and Lord of Lords” it is speaking of Heavenly - and not earthly - rulers?”

The scriptures do mention Heavenly kings; Paul wrote to Timothy:

If we go on enduring, we shall also rule together [with the Christ] as kings
2nd to Timothy 2:12

and John wrote in the Book of Revelation of

...twenty-four thrones [surrounding Almighty God] and seated upon these thrones were twenty-four elders dressed in white outer garments, and with golden crowns upon their heads
Revelation 4:4

and the Millennial Reign of Jesus Christ with righteous ones raised from the earth to kingship with him in the Heavens.

1st Timothy & Revelation
However, Jesus’ role as King of kings is not refering to these. The title “King of kings and Lord of lords” occurs in 1st Timothy and in Revelation, in each case in the context of human rulers.

Paul wrote of Pontius Pilate - the lord of Rome over Jerusalem - when he mentioned Christ as “King of those who rule as kings and Lord of those who rule as lords” (1st to Timothy 6:15), with no word of Heavenly souls.

However, most telling of all is the account by the apostle John in the Book of Revelation. There John describes:

...the kings of the entire inhabited earth, [gathered] together to the war of the great day of God the Almighty... to the place that is called in Hebrew: Armageddon.
Revelation 16:14, 16

John explains further

These have one thought, and so they give their power and authority to the wild beast. These will battle with the Lamb, but, because he is Lord of lords and King of kings, the Lamb will conquer them.
Revelation 17:13-14

Jesus could not vanquish these until he is given authority as King over them, which begins when the Divine rescue takes place. Thereafter, he has the authority in the earth to counter insurrection with a rod of iron when necessary (Revelation 19:11-16). If Jesus were King of Heavenly kings, then Revelation would not refer to them as “Kings of the earth”, and it would imply a rebellion either by those raised to Heavenly life - a kingship they do not possess until after Armageddon (Revelation 20:4).

Truth is Simple
Ever since the Christ opened the doorway to the Heavens for people of the earth, it has become tempting to apply all Biblical details to the Heavens. However Revelation is about the earth’s recovery, earthly kings in revolt, earthly people regaining a firm footing before Almighty God. Christ will indeed be the king of those elevated to Heavenly kingship for the final thousand years - the “Millennial Reign” - of his long custodianship of the earth, but both Paul and John were describing his authority over the revolt by earthly kings at Armageddon, the antithesis of the call to the Heavens for help at the end of our independent era.