There are multiple ways a nation could be lead under such circumstance.
One would certainly be a true democracy where each meaningful political decision is made by plebiscite. There would still need to be political institutions that prepare such plebiscites as well as executing them. Since no decision is made by a single person neither in preparation, decision or execution, this would avoid the loss of political leaders. It does require serious planning, education of the voters and time. In such a world it could be mandatory to have certain time a week dedicate for working through the plebiscites as otherwise the nation would not properly work. In order to complete this task using the Internet and computers or perhaps even a dedicated network for just that purpose could greatly improve efficiency.
Another way could be that in a fixed time interval e.g. 4 years a groups of citizens get randomly selected (computer based). They form the heads of the government organizations. They personally meet and work on their assigned areas (laws, taxes, military etc.) but voting takes place anonymously. In case somebody dies a new citizen gets randomly selected and takes the place.
If technology would have greatly advanced, specialized A.I. could also complete the job of governing a nation. Assuming that specialized means that they are just made for their distinct purpose to solve complex tasks, but are not human-like. If this would be possible and the systems are very reliable, decisions could be made very fast, adapt to situations right-away, optimize processes. Additionally they are not bribeable, not racist, not emotional and most importantly not killable.
In general the problems are not just for politics but for everyday life. So assuming that a somewhat functioning society and economy should exist, it might actually be a situation where partly ideas of communism might even work (Not my personal political opinion). One part that would be crucial due to possibly very spontaneous deaths and low life-span, that the right amount of people are educated in certain fields (e.g. medicine, agriculture, economy, law etc.) so that there will always be a person to take another persons place so that we do not run out of knowledge and skills. If a certain job has a specifically high death rate, more people are needed unless there is a way to avoid this while still being efficient at work. This could mean that each person is assigned to such a field at birth to make sure everything works.
As most things in life in that world would still need human interaction another usual problem would be envy. So if somebody would have a better live for some reason, like being rich, that would certainly increase the risk of death. So if everybody would get equal support by the state, taxes, medical support, payment etc. it would probably be better for everybody's health.
Some jobs are especially at risk like judges. This would mean we would also need to change a lot of organizational structure throughout society. For courts that could mean that every decision is made by a jury.
As you see there are plenty areas that would be affected by this. Perhaps humanity would learn with time, that using this ability affects everyone negatively. If people would use their ability to much it could potential threaten the survival of the human race.
I will update this answer if more solutions come to my mind.
