I let you know strictly ----- I never heard this sentence used in English. "I let you know soon" - it would be more correct to say "I will let you know soon". We do not use the present tense to indicate the future tense as is common in Italian.

Yes, it can mean that. I hear it more often in the sense, "I'll let you know as soon as I find out or as soon as I know". It's not a common expression.