The Kings were given the duties of priests and were given religious roles. The Kings were regarded as intermediaries between the gods and men. They were responsible for making sacrifices to the Gods at times such as before leaving on a campaign, or every month they offered a fully grown animal in sacrifice to Apollo. The kings were also responsible for leading the citizens in their religious celebrations.
Spartan Kings held limited Judicial roles yet still had the authority of judges in certain cases. They decided upon legal matters such as the marriage of orphaned heiresses and had control of all cases concerning highways, public infrastructure like roads and water supply. Any adoptions of Spartan sons could only be approved by Spartan Kings themselves.
Their only Kingly power was as commanders in battle. If they wanted to, they had the right to declare war without permission from others but in fact they never did act in this way. In time of war both Kings led the Equals out onto the battlefield together, however, this was no longer after two kings got into an argument with eachother which led to one of the Kings withdrawing his troops, thus ruining the success of the expedition. After that, a law was pased forbidding both the kings to be away from the city on a campaign at the same time.
As well as having only limited power, the kings could also be punishmed for mis condunct, or even deposed. A king could be fired if a battle was a failure, or if things in gernal were going wrong within the city, the King would be blamed. This was due to the fact that they, as the city's priests, were held responsible for keeping the Gods happy.