Actually, I don't completely understand what exactly you want to know. Are you confused about which is correct, "let you know" or "tell you"? You could say/write either of them, but not both together in one sentence.
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AnonymousRepeating someone's words sometimes sounds a little childish to me, in this case, B's 'let you know'. What would you think?Did you mean to say ... A's words? Since A speaks first in the conversation you've given. Well, that doesn't sound childish to me at all. But if you feel so, B could just say, "OK, I will", or "Okay" in response to