Are Shakespeare's works written in Middle English?

No.  The works of William Shakespeare are written in what
is known as Early Modern English.  Middle English was used between the late 11th and
late 15th centuries.  Shakespeare was born in 1564, well after the date of 1470 that is
usually given as the end of the era of Middle English.

A
major piece of literature that was written in Middle English was The
Canterbury Tales by Geoffrey Chaucer.  If you look at a little bit of that
work in the original, you will see how different it is from original Shakespeare
material.  Here is the beginning of Chaucer's work in the
original:

WHAN that Aprille with
his shoures soote

The droghte
of Marche hath perced to the roote,

And bathed every veyne in swich
licour,

Of which vertu engendred is the
flour;

Chaucer's Middle
English is essentially unintelligible to us whereas Shakespeare's language is archaic,
but basically understandable in the original.

Thus, it is
clear that Shakespeare is not writing in the same language as
Chaucer.