Part of why this didn’t happen in Germany has to do with the nature of the monarchy during the imperial period. Unlike in Italy, which had a constitutional and fairly weak monarchy, or Japan, with a comparatively stronger monarchy than Germany’s that could be relied upon to implement an ultranationalist program, the German monarchy was somewhere in between and therefore ill fit to be at the center of a fascist government.

The other factor, which helps to make more sense out of the first, is the important role of the leader in fascism (Führerprinzip). In Italy, a weak monarch would not stand in the way of Mussolini playing the role of Duce. In Japan, the Emperor could play the role of leader himself. In the case of Germany, the role of the leader was so important that anyone other than Hitler in that role was unthinkable. A second fiddle role was not one the Kaiser or any potential successor would be prepared to play. He had been substantially empowered during the imperial period, including control over the army, and Hitler couldn’t have that.

So essentially a fascist will only tolerate a monarch who can either be a figurehead for the movement and/or stand aside and largely cede any important roles to the fascist leader. This worked in Italy and Japan but not Germany.

Stanley Payne discusses some of the inherent contradictions between monarchism and fascism in his History of Fascism: 1914-1945.

In addition to @thamesdarwin's excellent response, I just add two things:

1. Almost the entire German officer corps had trained in the imperial army and had sworn allegiance to the Kaiser, remaining monarchist even after the war. After the end of the monarchy, their oath of allegiance had been dissolved and replaced by a new oath to the Republic. In 1934 this oath was changed again and became a personal oath to Adolf Hitler: "I swear to God by this sacred pledge that I will render unconditional obedience to the head of the German state and people, Adolf Hitler." This oath helped greatly to strengthen the regime's grip on the armed forces, making it morally difficult for officers to rebel against the dictator (military oaths are taken very seriously by those involved). Had the Kaiser returned, the oath would almost certainly have been turned back to his figure and, as happened in Italy in 1943, the armed forces could have been turned against the dictator as soon as his power weakened.

2. Hitler's formal role was "Führer und Reichskanzler," that is, head of state and head of government of the Reich. Thanks to these two positions Hitler could rule by circumventing the democratic mechanisms of the Weimar Republic, whose constitution formally was never abolished and the official name of the German state remained Deutsches Reich throughout the period 1919-1945. In the Weimar constitution (as in many current Western constitutions) the head of government could issue urgent decrees without going through parliament, provided they were countersigned by the head of state. Of course in Hitler's case he countersigned the decrees himself and thus had almost absolute power. In the case of a restoration of the monarchy, the head of state would have been the Kaiser by necessity. And so there would have been a dangerous (from the Nazi point of view) impediment to Hitler's will: it would have been enough for the Kaiser not to sign to make the implementation of the laws more cumbersome (but not impossible, with 99% of the seats in parliament in Nazi hands). Then clear, in Italy the dictator had the same theoretical problem and King Victor Emmanuel III at best mumbled but then signed everything. But, as explained above, the German monarchy was not the Italian monarchy and Wilhelm II was not Victor Emmanuel III.