Do you think that I could ever learn to speak . . . . . . . . Japanese . . . . . . . . way . . . . . . . . Japanese speak it?

We do not use articles in front of proper nouns. Since 'Japanese' is a proper noun (in the first case it refers to the language, the second time it refers to the people) we omit the article in the first blank.
When the proper noun refers to a group of people, like 'the Japanese' collectively refers to the citizens of Japan, we use the definite article 'the'.
The noun 'way' is specific as it expresses a particular manner in which the Japanese speak.