It seems like a King, believed to be the divinely appointed ruler of the earth, would be see themselves as above reproach or real sin. And what rank of priest would have the right to judge their acts and assign punishment? Only a peer could do such a thing I would think.
Medieval kings and emperors were very much obliged to go to confession and perform penance as needed. They were baptized and thus members of the Christian community bound to receive all the sacraments the laity received. They would confess to ordinary priests or bishops as they desired, though, just as other laity, if they committed certain sins, these could only be absolved by a bishop or the pope. The idea that kings were divine and exempt from human law—the “divine right of kings”—is a notion only forwarded in the 17th century. Medieval kings were bound both by human and divine law.
There are some very famous cases of penitent rulers. Ambrose, bishop of Milan, excommunicated the Emperor Theodosius I after he sacked Thessaloniki in 390 and killed several thousand people. He was forced to do penance for a few months. The two most famous examples of royal penance were Gregory VII’s excommunication of Holy Roman Emperor Henry IV in 1076. Henry had to petition the pope for forgiveness. He traveled to Canossa in northern Italy where Gregory was staying; he wore a penitential hair-shirt on the way and some say he walked part of the way barefooted. He stayed outside the castle in Canossa for three days before Gregory admitted him and absolved him. In 1170 over-eager knights in the retinue of Henry II of England murdered the Archbishop of Canterbury Thomas Becket in his cathedral. Henry had to make a penitential pilgrimage to Canterbury in 1174 where he publicly confessed his sins (an action somewhat rare by this time) and received symbolic lashes with a rod from the assembled bishops and the 80 monks connected to the cathedral.
King Louis IX (d. 1270), later made a saint, regularly confessed his sins and received symbolic lashes from a whip by his Dominican confessor.
Source: This is off the top of my head as a historian of medieval religion. I can give you some sources if you like.
If I could add to this question: was there any advantage to being the ruler's confessor? I seem to recall reading this in some fictional accounts of Versailles, where being the King or Queen's confessor would give that priest a great deal of political power...