Jezebel was not Jewish by birth (Kings 1:16:31), nor do the verses say she converted. How did her two sons become kings of Israel and why didn't anybody like Elijah or Elisha (and later on especially Ezra) comment on this apparent abomination?
Some Rishonim say explicitly that Jezebel did not convert, and that was part of Ahab's sin in marrying her (see, for instance, Ralbag to Kings 1:16:31), and R Shlomo Primo writes (Imrei Shefer, Vayera) that Joram was also not Jewish. While this may not have been explicitly mentioned in the verses, Jezebel's children, and indeed all of Ahab's decendants (it's not clear if he had children through other wives, see Yalkut Shimoni 232), were all hunted down and killed, so it's clear they weren't finding favor in God's eyes. It's likely they ruled by force, not based on any Halakhic decisors. (Note the question becomes even more complicated as Athalia, in the line of David, may be a descendant of Jezebel's (see Kings 2:8:27).)
However, Tosfot to AZ 26b says Joram was a Jew, and the Yerushalmi (Horyot 3:2) says that Jehu (who came later) was the first of the kings of the northern kingdom to not be Halakhically a king. They likely hold that Jezebel converted and then one has to analyze the laws of conversions to determine what would constitute sufficient acceptance of the Torah to qualify (perhaps she intended to keep the commandments for an initial period, or perhaps her intent to marry her husband binds her to his commitment to the commandments; see the articles above for details).
See Tzitz Eliezer 10:41:2 and Yabia Omer YD 5:13:2 who discuss possible legal applications of the different positions.