The translation problem in II Kings 5:10 is that what starts out as a declarative sentence ends in what most commentators read as an imperative form, "va'tahar" (וטהר), "be pure!", as in "be healthy!". This makes the verse sound a little clumsy in Hebrew, as if Elisha said:
"Go and wash in the Jordan seven times, and thy flesh shall come back to thee, and (by the way) be pure!"
In verse 13, Naaman's advisors quote or paraphrase Elisha as having said
"Wash and be pure!"
which sounds just fine as an imperative sentence. The "be pure" in verse 13 is the exact same word, "va'tahar" (וטהר) as in verse 10.
The English translation of verse 10 quoted in the OP,
"Go and wash in the Jordan seven times, and thy flesh shall come back to thee, and you will be cleansed"
"fixes" the clumsiness in the Hebrew by reading "ve'tit'har (ותטהר) instead of "va'tahar" (וטהר), adding, as it were a single consonant, "tav" to the text thereby changing an imperative to a future passive, "will be cleansed". It could be that the translator felt that this was the the correct spelling in the original text, or at least the original meaning of the text. Note that elision from one spelling or form of a word either back or forwards a few lines to another occurrence of a similar word is a common copyist error.
In any event, this is a relatively small issue that is noted by the modern commentators but does not warrant a comment by Rashi, who usually picks up on this type of problem.
[See the grammatical analysis in Yehuda Keil's commentary on II Kings in the "Daat Mikrah" series published by the Mossad HaRav Kook, 1989, Jerusalem (Heb.)]
The following suggests answers to the problem in the verse that is specific to the Rabbinic viewpoint, which is that non-Jews are not considered to be susceptible to ritual impurity or uncleanliness (called "tumah"), so how can Naaman, a non-Jew be impure? Although "tsaraat" or a similar condition, "sh'cheen" is said to afflict non-Israelites in Exodus 9:9 and in Job 2:7, in neither case is the concept of ritual impurity used.
The Rabbinic view is presented in the Mishna, order Purity (taharot), tractate afflictions (negaim), chapter 3:1. There is no Talmud, either Babylonian or Jerusalem on this tractate, so the reasoning for the Rabbinic ruling has been lost. However, the probable logic is reconstructed in the Mishne L'Melech commentary on Maimonides's Mishne Torah, order Purity, section Leprosy (tsaraot), chapter 9. I will spare you a recap here.
So, there are at least five possible answers, the first two of which should be acceptable to the Rabbinic viewpoint, and the last three probably not:
Although the text in verses 10 and 13 uses the Hebrew word for ritual purity, "taharah", the Aramaic translation by Jonathan ben Uziel uses the word "itasei", to be healed, in both instances. In Lev 13, on which the Mishna is based, in both of the Aramaic translations that we have, that of Jonathan ben Uziel and that of Onkelus the Convert, the Hebrew "taharah" is translated using the corresponding Aramaic word for purity "dacuyei". The Rabbinic commentators who note this use of "itasei" in II Kings 5 imply that although the Hebrew text here uses the word for ritual purity, the intent in this instance is in fact healing.
According to the extra-biblical oral tradition, based on Naaman's confession of faith in II Kings 5:15, Naaman became a convert to the Israelite faith at the time of his immersion in the Jordan river [Rif]. In this case, as in the case of any convert to Judaism to this day, the newly converted is in a state of tahara ("ritual purity") following his immersion. If Elisha foresaw this outcome by prophetic vision, then verse 10 makes sense even if Elisha held that non-Israelites are not susceptible to tumah.
Since the time of the Mishna at least Jewish tradition does not consider non-Jews to be susceptible to "ritual impurity", i,e, tumah, with the exception of tumat-met, the tumah conveyed by a corpse. It could be that before the exile there was another tradition that Elisha followed according to which Lev. 13:2 was taken more literally and "adam" (first word of the verse) was read to mean any human, whether Israelite or not, or it could be that Elisha is making a statement of position in the ongoing universalist versus nationalist polemic in Israelite theology.
The concept of tsaraat (translated as leprosy) conveying a state of at least ritual impurity was not unique to the Israelite community. So Elisha's statement to Naaman about being purified from his condition could be phrased in Naaman's idiom of purity from tsaraat. See Naaman's own statement in verse 12.
The naive hypotheses - that the entire story of Naaman is told from within the cultural frame of reference of the Israelite community, as if Naaman were Israelite, and the actual inapplicability of some of the concepts of ritual purity and impurity to him as a non-Israelite is simply ignored.
The word "taharah" in the OT is not connected to modern ideas of physical cleanliness. The common English translations render it as "cleanliness", but it is cleanliness in a ritual or spiritual sense.
The medieval commentators (students of the Rashi) ask this question too, the ancient commentators (Hazal) do not. Unfortunately, neither provide much in the way of answers. There is nothing on this question in any of the usual commentators of the "Mikraot Gedolot" editions of the Hebrew Bible - Rashi, Radak, Metsudat David, Metsudat Tsion, or Ralbag, nor in the midrash Yalkut Shimoni, nor in the Abarbanel. The Rif (usually published together with the Abarbanel) provides a beautiful, spiritual explanation of Naaman's leprosy and of the meaning of going down to the Jordan river but does not explain how the impurity of leprosy could apply to Naaman in the first place.