It is very difficult to come up with a clear length of the ministry from the data in the NT and Josephus, as there are many different conflicting facts (such as the year when Jesus was born or the age at which Jesus was when he died). If I were to follow just the data found in Luke and Josephus, there could only be a year in Jesus' ministry. Luke dates Jesus birth to the census of Quirinius in AD 6 (2:1-2, that this census occurred in AD 6 is confirmed by the second mention of it in Acts 5:37), and he dates the start of John the Baptist's ministry to the 15th year of Tiberius Caesar (3:1-2), i.e. AD 29. The Society and many others take this to be the date of Jesus' baptism, but that is not what the text says. Since Luke has John the Baptist born around the time when Jesus was born, this means that John was about 23 when he started his ministry. We know that John had a very large movement by the time he died, considering what Josephus said about him. We later learn in v. 23 that "Jesus himself was almost thirty years old when he began his ministry", i.e. 29, and since John was about the same age, this means that John had about 6 years to build his ministry. Then Jesus was baptised and Jesus started his ministry when the latter was 29, i.e. in AD 35. Although the other gospels say that John baptised Jesus, this is not what Luke says: it doesn't state who the baptiser was and it says that John was already imprisoned by Herod (3:19-22). Next we see that while John was imprisoned, Jesus arose in his place as a successor in the ministry, with Jesus recognizing the "baptism of John" and with John showing interest in Jesus' work and by having his own messengers sent to see who this person was preaching in the wilderness (7:18-35). Then at some point, Herod beheads John (9:7-9); Luke does not narrate the execution itself but only mentions it after the fact and makes clear that this occurred around the time Jesus was becoming famous as a successor of John (the same author in Acts 18:24-9:6 describes how John's disciples later became Christians). We do know from Josephus roughly when this happened: the defeat of Herod's army in AD 36 was popularly regarded as divine punishment for Herod's execution of John (Antiquitates 18.116-119), and this battle occurred because Herod had divorced Phaselis the daughter of the king of the Nabataeans to take as his wife Herodias, the wife of his half-brother Philip. The account of the beheading in Mark also links the death of John to his criticism of Herod's marriage to Herodias (6:17-28), which had occurred after Philip died in the 20th year of Tiberius, i.e. AD 34. So this dates the execution of John pretty securely to sometime between AD 34 and 36, and if the beheading occurred on Herod's birthday, then the date would be August of AD 35. This fits perfectly with the date above re the start of Jesus' ministry in AD 35. Note that Luke does not depict John languishing in prison for years and years; it depicts the time between Jesus' baptism and John's execution as rather brief. And then Jesus is construed as himself executed the following Passover, under Pontius Pilate and high priest Caiaphas. This too fits with what Josephus (Antiquitates 18.88-90) writes about Pilate: Pilate was deposed in very late AD 36 or early 37 just before the Passover and shortly before the death of Tiberius Caesar on March 16, AD 37. And Caiaphas was also deprived of the high priesthood on Passover the time when Tiberius Caesar died (18.95), i.e. March-April AD 37, so Jesus could not have been executed as late as AD 37. So if you just go with the data in Luke-Acts and Josephus (and perhaps the story of John's execution in Mark), the picture is that of Jesus' ministry spanning about a year between AD 35 and 36.
This of course strictly ignores the very different traditions found elsewhere, such as in Matthew (which places the date of Jesus' birth much earlier) and John (which has a very different chronological framework). But it also makes sense to read Luke-Acts with Josephus since the former is possibly dependent on the latter as a source. Note too that starting Jesus' ministry as early as AD 29 creates huge problems with Jewish history: Herodias was still married to Philip and not Herod as late as AD 34 (and Herod was still married to Phaselis), John the Baptist was executed rather close to AD 36 and probably not many years earlier, etc.