The traditional answer to reconcile Shemot 21:6 and Vayikra 25:10 is that when the Torah uses the word le-olam in Shemot 21:6 and Devarim 15:17, the meaning is until the yovel year because fifty years is a long period of time (Talmud Kiddushin 21b, quoted by Rashi on 21:6 and on Devarim 15:17). Yet, this is not the meaning of the word le-olam, and the Rashbam (12th century, France, on Shemot 21:6, also see Rashba, towards the end of Responsa 9) writes that the simple reading of 21:6 is that a slave could choose to remain a slave forever. Furthermore, when a slave chose to be a slave "forever" a hole was bored in his ear, which Rashbam (on 21:6) explains was a sign that the person was a slave. This sign only makes sense, if the person was a slave until he died, as otherwise he would have a mark that he was a slave when he would be free.
The Rashbam is following his revolutionary idea that he states in his introduction to parashat Mishpatim, that he intends to explain the simple reading of the Torah, even when it does not accord with the halachik understanding of the Torah. To the best of my knowledge, this change only began with the Rashbam, as for instance, Rashi and Ibn Ezra do not offer explanations that counter the halachik interpretations. Prior to the Rashbam, there were many opinions that one does not have to follow Midrashim in reference to non-halachik texts, but not to follow the halachik interpretations is potentially more dangerous since people can argue that one can act based on the person’s understanding of the text. To counter this possibility, the Rashbam in his introduction to parashat Mishpatim, writes that the primary interpretation is the halachik interpretation. Touitou (2005, pp. 177-188) suggests that the Rashbam developed his new approach due to arguments with Christians in 12th century France. While many people do not accept the Rashbam’s approach, this blog (commentary) adheres to it.
Hoffmann (1953, p. 236, on Vayikra 25:40) notes that the Gra (1727-1797) wrote that this case is one of the instances where the legal tradition uproots the simple reading of the text. The idea being that we follow the tradition that a Jewish slave cannot be a permanent slave even though the simple reading of the Torah is that a slave could choose to remain a slave forever. Yet, while this rule tells us how to practice, still the contradiction would remain for understanding the Torah, as Shemot 21 and Devarim 15 imply that it is possible for a Jewish slave to be a slave forever while Vayikra 25 rules out this possibility as all Jewish slaves became free during the yovel year. I did not see that the Rashbam addressed this question.
Hoffmann (1953, pp. 236, 237) suggests that the meaning of le-olam in Shemot 21:6 and Devarim 15:17 is that the slave can never claim his freedom from his owner. However, during the yovel year, all slavery is ended without the slave having to claim his freedom and hence even a slave who cannot claim his freedom goes free in the yovel due to the national abolishment of slavery.
Barry Eichler (1985) seems to suggest that the laws of Shemot and Devarim refer to slaves who are sold because of debt, and these slaves could remain forever slaves, while the laws of Vayikra refers to slaves who choose to be slaves because of poverty, and these slaves cannot remain as slaves forever. I am not sure if this distinction between the types of slaves in the different chapters accords with the simple reading of the Torah since the Torah does not state or even imply that the slave being referred to in Shemot 21:4-6 was sold into slavery due to debt. Also, while Vayikra 25 refers to a person who is so poor that he/ she sells himself to be a slave, these passages (Vayikra 25:39-44) are after the Torah stated that all slaves go free in the yovel year (Vayikra 25:10).
I would suggest that when the slave declares that he wants to stay a slave, then he has annulled the possibility of going free in the yovel year. The reason is that Vayikra 25:10 states that all slaves are to be freed to return to their ancestral property and to their family. However, by the case of the slave who wants to remain a slave for more than six years, he is stating that his family is the wife the master gave him, his children, and/ or the master. This statement by the slave means that he is legally replacing his biological family from his birth and his claim to the ancestral property with his new family, the wife, his biological children or the master. Thus, the slave does not go free in the yovel years since in his case to return to his family means to stay a slave.
Note the word kol, all, in Vayikra 25:10, does not mean literally all, but a majority, and this is the definition of the word kol in many verses in the Torah.
Bibliography:
Eichler, Barry, 1985, On slavery, Harper's Bible Dictionary. edited by Paul J. Achtemeier, San Francisco: Harper & Row, p. 959.
Hoffmann, David (1843-1921), 1953, Leviticus, Jerusalem: Mossad Harav Kook.
Touitou, Elazar, 2005, Exegesis in perpetual motion: Studies on the Pentateuchal commentary of Rabbi Samuel ben Meir, Ramat Gan: Bar-Ilan University Press.