The victims appeared mainly to have been Afghan civilians.
German Foreign Minister Sigmar Gabriel said the "despicable" attack killed an Afghan guard from the German embassy, and added that some employees had been injured, though he did not give further details. Afghanistan's private TOLO Television also reported a staffer killed; Germany said an Afghan security guard outside its embassy was among those killed. A spokeswoman for the German foreign ministry said the bomber's target was unknown.
"Let's be clear: This is an intelligence failure, as has been the case with so many other attacks in Kabul and beyond".
German officials said they were constantly reviewing security in Afghanistan.
"We are all safe, all our staff, all our personnel are safe".
The first report said 19 people had been killed but the death toll rose rapidly as ambulances and police vehicles rushed the wounded and the barely alive to hospitals.
No group immediately claimed responsibility for the blast.
The blast, which shattered windows and blew doors off their hinges in houses hundreds of metres away, was unusually strong.
Afghans were the second largest group of asylum seekers in Germany after Syrians a year ago and four months before an election, Merkel wants to make sure voters think she has the migrant crisis under control. The bombing killed 58 people and wounded 141.
A truck bomb exploded in the embassy and presidential palace quarter in Kabul during the heart of morning rush hour, killing scores and injuring hundreds in one of the most heavily secured areas of the city.
The Taliban have been stepping up their push to defeat the US-backed government.
The US has about 8,400 troops in Afghanistan. Ashraf Ghani's government, however, holds all provincial centres.
Afghan troops are backed by U.S. and North Atlantic Treaty Organisation forces, and the Pentagon has reportedly asked the White House to send thousands more soldiers to break the deadlock in the battle against the Taliban.
The commander of USA forces in Afghanistan, General John Nicholson, told a congressional hearing this year that he needed several thousand more troops to help Afghan forces break a "stalemate" with the Taliban.