From what I think is that once there was subsequential decrease in ninjas during the war and that after Minato successfully ended the long war that started with the 3 Sannins, Jiraiya, Sarutobi and the elders all decided that he rightfully deserve to become Hokage.
In Kakashi's Gaiden, it's seen that he was only refered as Yellow Flash and not Yondaime Hokage. Now, kakashi didn't seem to have much grown up when Kyuubi attacked, so it must be just after the 3rd war.
I'll edit later perhaps with some more calculation (I made a post something like that with calculations, if only I can find it again)
EDIT: Found it
.:KingScorp said:
Okay, let me also analyse it...
When Naruto was 12, Tsunade was said to be 50.
Hence, Jiraiya was also 50...
Now when Naruto is 16, Jiraiya must have had 54
16 Years ago, when Kyuubi attacked and Minato died, Jiraiya must have had 38 Years old...
Minato was placed under Jiraiya's supervision at 10 years old and Minato surely took charge of Kakashi (then around 13) at around 17 years old...
Let suppose now that he took about 3 Years to win the 3rd War and to create Rasengan, some months to pass on the legacy of Rasengan to others and finally, 9 months till the birth of Naruto and his death.
So when Minato died, he should have approximately 17+4 = 21 Years old.
Okay, this weren't calculations for Minato but for Jiraiya but I think it's good. So, there we are. He surely became Hokage when he was around 18 years old and died during his reign at 22 (or nearly so...)
So that makes up that he was Hokage for 4 years or so...
Kakashi, being then 17 makes sense at Minato's death as 16 years later (Now, he should have around 33 which is understandable since his age is completely defined)
Now, perhaps, he also created the Rasengan before the war, then that will make up to 2 years of reign as Hokage. (thus Kakashi now=31 years ---> more understandable) This 2 options pretty much will fit in the timeline