In medieval England, the King's peace was simply the general peace provided to the kingdom as a whole by the law administered in the King's name. From this, it is clear that "the King" in this context was simply the ruling monarch (variously Henry VI, Edward IV, Richard III, or Henry VII).
In fact, we still have the concept of the Queen's peace in modern British law, and the offence 'committing a breach of the peace' is disorderly conduct that breaches the Queen's peace.
The meaning of the quote becomes more clear if we look at it in the context of the whole paragraph from which it is taken in Russell's Political Ideals:
In the relations between states, as in the relations of groups within a single state, what is to be desired is independence for each as regards internal affairs, and law rather than private force as regards external affairs. But as regards groups within a state, it is internal independence that must be emphasized, since that is what is lacking; subjection to law has been secured, on the whole, since the end of the Middle Ages. In the relations between states, on the contrary, it is law and a central government that are lacking, since independence exists for external as for internal affairs. The stage we have reached in the affairs of Europe corresponds to the stage
reached in our internal affairs during the Wars of the Roses, when turbulent barons frustrated the attempt to make them keep the king's peace. Thus, although the goal is the same in the two cases, the steps to be taken in order to achieve it are quite different.
We can see that he is simply comparing the absence of law and government in relations between European states in 1917 with the not dissimilar situation that existed in England during the Wars of the Roses, where a breakdown of law and governance was commonplace. There is absolutely no suggestion here that Russel was referring to Henry's mental issues.