The only one who is allowed to pick a new Hokage is the previous Hokage himself/herself. If the former Hokage dies or is incapacitated before being able to do so, the Fire Daimyō will appoint a Hokage Candidate upon the recommendation of the Konoha council and the Fire Daimyo council. Afterwards, the candidate will need to be approved by the jonin to become officially Hokage.
If Tsunade died instead of falling into a coma, and picked Danzo herself, Danzo would have been Hokage, no questions asked. But thats not the case. The Fire Daimyo appointed Danzo, but the jonin never did, so he was not an official Hokage.