The gemara in ברכות י"ב states that during the ten days of repentence we change the nusach of the שמונה עשרה and say המלך הקדוש instead of האל הקדוש as well as saying המלך המשפט instead of מלך אוהב צדקה ומשפט. Rashi on the gemara comments that המלך המשפט is grammatically incorrect, it should be מלך המשפט and it is to be understood that way (the king of משפט) and basically we ignore the extra ה. The Beis Yosef comments that the same problem should apply to המלך הקדוש and yet Rashi doesn't say anything. He quotes some who say that Rashi understood that המלך הקדוש should be understood as 2 separate titles, the translation would be "the king, the holy one". The standard translation is "the holy king" (the 2 words are 1 phrase) like המלך המשפט the king of משפט.
Interestingly enough this is a מחלוקת Artscroll and Metzudah. Artscroll in their siddurim and machzorim translate it as "the holy king" while Metzuda translates it like this interpretation of Rashi "the king, the holy one".
To sum up there are 2 interpretations of המלך הקדוש
1. the king, the holy one (which is the literal translation with the ה at the beginning of the word)
2. the holy king (the 2 words are 1 phrase) like המלך המשפט the king of משפט