A noun with a plural form that's the same as its singular form does not make it an uncountable noun. No English speaker would tell you that "sheep" is uncountable even though its plural form is still "sheep."
Kanji (and its plural form, kanji) is definitely a countable noun in English. I don't know what makes you think that Japanese nouns are uncountable in either Japanese or English, but that's not true at all.
EDIT: One could argue that "kanji," the concept of Chinese characters, is an uncountable noun, and that in order to classify it in English, one would have to say "characters of kanji." In this usage, I can agree that "kanji" is an uncountable noun. However, in the case where it's used as the OP said such as "I know about 10 kanji," where "kanji" is used as a gloss to mean the individual characters in the Chinese character set, it seems evident that it would be a countable noun.