But this means the government can literally get away with murder.
I suppose you are thinking about the government ordering someone to be murdered or committing murders themselves?
Well, it's still illegal to kill someone, it will of course be up to the courts to decide whether you are to be punished for killing someone or not, like in self defense.
It is also covered by §71 in the Constitution, which says that the personal freedom of Danish citizens is inviolable. You cannot be imprisoned in any form due to among other things your political convictions. - I'd say killing political opponents is a violation of §71.
- On top of that no one are obliged to obey an illegal order, in fact you risk punishment at some point if you do. So being ordered to kill someone by the government, can and should be refused.
As for MPs killing someone. If the police investigation points to an MP the Parliament can lift the immunity and considering the severity of the crime, no doubt will.
Beyond that point it begins to become really serious and complicated!
Because according to §13 of the Constitution King Frederik is free from responsibility for the conduct of his government. The government ministers are responsible for misconduct or breaches of the law. Even if King Frederik has signed such a law, which he BTW is obliged to.
But... In theory King Frederik can veto a law by refusing to sign, say a law that forbids socialists from crossing roads and streets outside traffic lights and pedestrian crossings. It shouldn't happen because all bills are vetted by legal experts before even being put up for a vote in the Parliament to see if it conflicts with another law or is in violation of the Constitution.
So, again in theory King Frederik is one of the last guarantees against a totalitarian government in DK.
Because
all power enacted or enforced by the authorities in DK is in his name.
According to §16 of the Constitution King Frederik can also by himself prosecute a government minster for misconduct.
In §14 it is King Frederik who appoints or dismiss a governments - they are his servants. Again in theory, King Frederik can dismiss a totalitarian government. - And should a government begin to breach the Constitution it is of course an open question whether King Frederik is obligated to step in. But that is way beyond my knowledge and something I can only speculate about.
And to make sure a rogue government doesn't arrest or dismiss King Frederik, we go back to §13. The person of the King is sacrosanct. And by extension his immediate family .
Not to mention that King Frederik has the right to put up bills for the Parliament to debate and vote on.
And he can act by decree if the Parliament is "out of order" so to speak. The decrees needs to be ratified by the Parliament later though.
To sum up, the Danish monarchs have a surprising amount of theoretical power - just in case...