There are technical names of the mixed phrases used in your first example, and I can't recall them at the moment, but I believe that it is grammatically incorrect. Let's break it into two parts:
- Please let me know
- what is the plan.
The first part is essentially a command (albeit a polite one), while the second phrase is a question. Questions in English are always followed by a question mark (?). However, your intended meaning is clear enough that this is NOT a question, but a request. I often hear this phrasing from several of my non-native English speaking friends, so you're definitely not alone in this regard.
A good rule to follow is that the interrogative "what" followed by a conjugate of "to be" is always going to be the beginning of a question (I can't think of an example where this doesn't hold true). When these words are separated by a pronoun or noun, it is no longer a question, and instead becomes a statement, e.g. "let me know what THE PLAN is".