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I think it has to do with what you mean by a “kingdom of hell”.

There seems to be a popular idea of hell as a place in which Satan is a ruler as God is a ruler in heaven (most recently expressed in the TV show, Reaper). I don't find that idea expressed in scripture. Presumably he is the one Paul referred to as the ruler of the powers of this dark world, the ruler of the kingdom of the air (Eph. 2:2), whatever that is, a spiritual force of evil in heavenly places (Eph. 6:12). But as far as words that are translated as hell in the NT, there is Hades (Matthew 11:23), there is Gehenna (James 3:6) , there is Tartarus (2 Peter 2:4), there is the Abyss (Rev. 20:1) and there will be a lake of fire into which Satan and those who have sided with him will be tossed (Rev. 20:10). But none of these five is described as a kingdom over which Satan rules, his demons torturing lost souls for ever and ever, as though that was a job provided for them to do by God.