I'll tell you what let's do

This is very colloquial, at best, if not downright incorrect. However, it is currently in use, as shows this ngram.

_{(Following copy of ngram mentioned above added as a later addition to make up for user psmears' problem)}

Syntactically, this construction is not analysable. It means "I'll tell you what we'll do.", which is used much more often: ngram.

The illogicality comes from the fact that a nominal clause must be replaceable by a noun phrase: it is the object of "tell", and therefore must represent a thing, whether a concrete thing or an abstract one; instead of a verb form telling what action is performed on the object "what" (in "what let's do") there is an order. That is not possible.

_{(Following reference below is a later addition to make up for user psmears' problem)}

(ref. 2020) “Now let's talk about 'clean,' son. Clean means no messes around—no paper, strings, bones, sticks, or anything that messes up the place. I'll tell you what let's do. Let's just clean up half the yard right now and look at the difference.

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This final addition of references below are just a few of those that can be collected.

(ref. 1, 1999) MISS NORTHERN : I seen them two or three times . Don't look at the feet . Let's don't look at the feet . JUDGE TODD : I tell you what let's do . MISS NORTHERN : Don't look at the feet . JUDGE TODD : Let's don't look at the feet .

(ref. 2, 1960) Good boy ! Now you've got them . Now we GO ! Still not enough gas ? Boy , how ough can it get . It's enough to make an ambitious guy discouraged . Tell you what let's do . Lower the density altitude and save a little gas that way

(ref. 3, 1965) Tell you what let's do . Let's clean off a nice leaf of that giant sagittaria . Then you lay a big batch of eggs .

(ref. 4, 1931) ( Synge , The Well of the Saints , Act III ) . The let - form often occurs in a substantive relative clause : ' Ah'll ( I'll ) tell you what let's do , Miss Leighton ! ' ( W. D. Howells , A Hazard of New Fortunes , II , Ch . II ) . ' I tell you what let's do : let's all run

(ref. 5, 1940) But if you feel that way about it , I tell you what let's do . I was going over today to the new chute we're putting in on Forty .

(ref. 6, 1997) I went over to the ones that were in the union and I said , “ Look , Eli needs that letter so I'll tell you what let's do . I'll memorize the first two lines of it and go in the bathroom and write it down .

(ref. 7, 1931) I'll tell you what let's do . Let's divide this one . I'll dance the first half with you , Robert , and the other half with you , Richard .

(ref. 8, 1985) I'll tell you what let's do . I would like for you to write me something , anything , about a time when you had a romantic attitude . You were really expecting something to be great ,

(ref. 9, 2011) Look , I tell you what let's do . I've got an idea . ” An hour later , mid - afternoon , the five of them were at a beach near Palmilla , just south of Ashley's favorite beach bar . The sky was a cloudless cerulean and the surf impressive .

(ref. 10, 1961) “Just look at your hair—all mussed up. I'll tell you what let's do. Let's cut it off! I'll do it for you—now, right now

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.

To answer the questions:

"I'll tell you what let's do." - What meaning of let's is used here

It is the same meaning that it always has - "let us". It translates roughly as "may it be the case that" and includes an unspoken appeal for agreement.

and

what is happening grammatically?

Grammatically, it is a complete mess. Although I suspect that the use by Major General Stanislaw Sosabowski is simply an example of bad English, other posts claim it is a colloquialism.

In the normal use of "I tell you what...", what is a pronoun = the thing that. ("I [will] tell you what (the thing [that]) we should do [[is] - we should attack...]").

It therefore follows that whatever follows "I tell you what..." should be a content clause describing what the thing is.

"Let's do" is not a content clause.