I'll tell you what let's do

There's a relatively common saying, used by at least some speakers of modern English:

I'll tell you what let's do.

What meaning of let's is used here and what is happening grammatically? It doesn't seem like an imperative.

You can find many instances of this from published books. Many from this millenium. See here for details.

It has been used in Futurama, To Kill a Mocking Bird, and famously by Gene Hackman's character, Major General Stanislaw Sosabowski, in the film A Bridge too Far:

Doesn't matter what it was. When one man says to another, "I know what let's do today, let's play the war game."... everybody dies.

There are other interesting posts regarding let's in subdialects of English, such as Is it "Don't let's" or "Let's don't"? (see all answers there). Those questions, though related don't begin to answer the question here.

It's obvious from context it means "I know what I/we should do". A mashing together or mangling of other more identifiable elements "I know what", and "let's do this..." The exact origin seems obscure, as is often the case with colloquial speech, and attempts to parse it are likely doomed.

It's a slightly dated expression: although you still see it in the 21st century, most examples are from the early to mid 20th century, and more recent uses are either for period flavor or comedy. (It appears to be a favorite of MST3K and Futurama writers, for instance, who appreciate its incongruity.)

Even back in the day, it appears to have generally been used by uneducated people or in casual speech, rather than being considered good formal English.

A good example giving some of the flavor is a 2003 interview with Geoffrey Wolff about his biography of the Pennsylvania-born author John O'Hara (1905-1970). Wolff, channeling O'Hara, says "All of us sit around and say, at some point in our lives, 'I know what let’s do.' And the rest is this wonderfully comic cataclysmic outcome of that proposition — 'I know what I’m going to do.'"

Wolff's voice is described by the interviewer as "as if we were on some country weekend, in the mid-1930s, before the start of World War II. We are sitting in comfortable lawn chairs, and the grass is perfect, and the drinks mildly chilled and you are telling me about this man O’Hara."

That seems pretty much the sort of occasion you might say "I know what let's do." With a friend, after a few drinks.

One of the more recent usages I found was from the New York Times, August 21, 2011, where someone is interviewed about a bad production of Porgy and Bess in Massachusetts. The speaker, from New York, is quoted as saying "Hey, kids, I know what let's do, let's put on 'Hamlet,' and Shakespeare surely would have put on a happy ending, had he lived longer, and yes, next we'll do 'La Traviata,' where she is cured, and yes, another happy ending." This is obviously meant to be a parody of the thought processes of someone out of touch, a rube or simpleton.

We find it for instance in a comic novel by Sinclair Lewis from 1915, where a character with dreams of showbiz stardom says "Say, I know what let's do--let's get up a swell act and get on the Peanut Circuit. We'd hit Broadway with a noise like seventeen marine bands". (There are two examples about theater, but I think that might be a coincidence.)

It doesn't appear to be much earlier than the late 19th century: Google Books has an example from a 1888 children's novel. So it's a dated, early 20th century colloquialism (mainly east coast USA) that some people still find amusing.