If Satan is frozen in Cocytus, who is in charge of Dante's version of hell?

The Roman poet Virgil is Dante's guide on the journey through hell. Virgil is able to do this because he is not a Christian and so has been condemned to spend eternity in hell. Virgil has gone to hell not because he is a terrible person, which he is not, but because he was born before the birth of Christ. Therefore, he was unable to know about and benefit from Christian salvation. He is a wise guide through hell because he is so rational, but he doesn't fully understand the importance—the eternal significance—of Christ's coming, describing him only as a "mighty Lord."
Dante probably chose Virgil as his guide through hell because in Virgil's epic poem the Aeneid, he refers to a descent into hell: Virgil, on multiple levels, is therefore a knowledgeable guide to the underworld.
If we think about it, Satan, the prince of lies, would be a particularly untrustworthy guide through the underworld, providing Dante with a warped picture of it. Dante is lucky to have the reasonable Virgil who will tell him the truth about what he is seeing; though Virgil, unable to leave hell, can't guide him to Purgatory or Paradise.