One opinion expressed in the Gemara (and mentioned in Rashi on this verse) explains that the Egyptian King did not actually die. Rather, he became a Metzora which are considered as if dead, and became a “new” king after recovery.
To understand this concept, refer to Koheles 8,8, ואין שלטון ביום במות (and there is no kingship in the day of one’s death). In the Medrash, Rabbi Levi states that close to 52 times it is written King Dovid. Yet, when Dovid was on his deathbed, it is written ויקרבו ימי דוד (and not King Dovid). That when one is close to death, one is no longer called “King”, since rulership does not exist on the day of one’s death.
Therefore, how could verse state that the Egyptian King died (should have stated that Pharoh died)? Thus, the verse must be referring to the King continued to be King but was in a state that resulted in a temporary vacancy and then resumed the Kingship as if new.
