If William Shakespeare wrote in broken English grammar, why is he so famous for his literary works?
I'm not sure what you mean. Most of Shakespeare's sentences are grammatical. Since he was chiefly a playwright, he occasional wrote sentence fragments, because that's how people often talk, and he was trying to mimic human speech. If anything, his writing was unrealistic in that his characters were much more often grammatical than people are when they're speaking in real life. 

Here's a typical speech: 

O, what a noble mind is here o'erthrown!

The courtier's, soldier's, scholar's, eye, tongue, sword;

The expectancy and rose of the fair state,

The glass of fashion and the mould of form,

The observed of all observers, quite, quite down!

And I, of ladies most deject and wretched,

That suck'd the honey of his music vows,

Now see that noble and most sovereign reason,

Like sweet bells jangled, out of tune and harsh;

That unmatch'd form and feature of blown youth

Blasted with ecstasy. O, woe is me,

To have seen what I have seen, see what I see!

-- Ophelia from "Hamlet."

Let's examine it from with our focus on grammar.

1. "O, what a noble mind is here o'erthrown!" is a perfectly grammatical sentence, with a subject ("mind") and a predicate ("is here o'erthrown").

2. The next sentence is more complex, but it's equally grammatical: 

SUBJECT:

The courtier's, soldier's, scholar's, eye, tongue, sword;

The expectancy and rose of the fair state,

The glass of fashion and the mould of form,

The observed of all observers, 

PREDICATE:

quite, quite down!

Well, I suppose you can quibble that it should be ...

"is quite, quite down!"

... but that would ruin the poetic meter, and it's unnecessary for comprehension, especially when spoken by an actor. It's hearing someone sigh and say, "My life ... totally ruined by drugs and alcohol!" There's no need for an "is."

3. And I, of ladies most deject and wretched,

That suck'd the honey of his music vows,

Now see that noble and most sovereign reason,

Like sweet bells jangled, out of tune and harsh;[and]

That unmatch'd form and feature of blown youth

Blasted with ecstasy. 

The above is perfectly grammatical, though complex. I have bolded the main structural elements of the sentence, which you could shorten to "And I now see that reason and that form and feature blasted," which, while confusing out of context, is grammatically the same as "And I now see that sofa and that table and chair burning."

The non-bolded words are gloss—modifying phrases. You could put them all in parentheses if you wanted. 

My addition of "[and]" isn't a cheat. I just did it to clarify. The semicolon after "harsh" serves the same purpose. Compare "I like cats and I like dogs" with "I like cats; I like dogs."

My high school English teacher would have complained about starting a sentence with "And," but that's a silly and arbitrary rule which writers have ignored for centuries. Even God (or at least the minions of King James I) ignored it:

In the beginning God created the heaven and the earth.

And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.

And God said, Let there be light: and there was light.

-- "The King James Bible," "Genesis"

Here, F. Scott Fitzgerald shockingly starts one sentence with an "And" and another with a "But."

I think he was afraid they would dart down a side street and out of his life forever.But they didn't. And we all took the less explicable step of engaging the parlor of a suite in the Plaza Hotel.

-- "The Great Gatsby."

Here's Jane Austen flagrantly ignoring the rule:

The happiness anticipated by Catherine and Lydia depended less on any single event, or any particular person, for though they each, like Elizabeth, meant to dance half the evening with Mr. Wickham, he was by no means the only partner who could satisfy them, and a ball was, at any rate, a ball. And even Mary could assure her family that she had no disinclination for it.

-- "Pride and Prejudice."

And Charles Dickens wouldn't give a sixpence for my English teacher's silly strictures:

So Mr. Kenge gave me his arm and we went round the corner, under a colonnade, and in at a side door. And so we came, along a passage, into a comfortable sort of room where a young lady and a young gentleman were standing near a great, loud-roaring fire.

-- "Bleak House."

4. "O, woe is me, to have seen what I have seen, see what I see" is a fairly simple, grammatical sentence, though to please your English teacher, you might want to change the comma between "seen" and "see" to a semicolon.

As to why Shakespeare is so famous, I doubt it has much to do with whether or not he's perfectly grammatical. As Elvis said, "You ain't nothin' but a hound dog!" By which I assume he meant bad grammar ain't gonna keep you off the charts. 

Shakespeare is famous at least partly because he used language better than almost anyone else. And I'm not talking about grammar. I'm talking about imagery, musicality, metaphor, meter, and surprise. He painted with words. He made words sing. 

He also did this over 400 years ago, so his English was a bit different from contemporary English. It takes some training for a newbie to get into it. Here are some posts that might help:

Marcus Geduld's answer to What are the secrets to understanding the themes and language of Shakespeare?

Marcus Geduld's answer to What is so great about Shakespeare?

Marcus Geduld's answer to Is Seinfeld at the level of Shakespeare in terms of comedy?