'Kindly make sure to let's know'

While "let's" is derived from contracting "let us", it is now a fixed term with a specific meaning. It would be slightly odd to write "let us" to mean "we should" (it's a bit overly formal), and it's very odd to write "let's" to mean "let us".

To be idiomatic, a contraction has be one that is standard, and being a standard contraction is a property of not only the words being contracted, but the meaning those words are being used in. So, for instance, if we use the word "will" in the sense of "last testament", it would not be idiomatic to say "Is the won't here?" to mean "Is the will not here?"