Is it okay to answer "Please let me know", short (without "when...", "if...", "what...", etc.)?
Consider for instance
-- I can check that for you tomorrow morning.
-- Yes, please let me know.
This is a perfectly fine way to respond to that statement. Adding an explicit time frame ("Please let me know as soon as you find out") is permissible and may be clearer, but it's not required.
You can shorten it further:
This is similar to sentences such as: