To repeat, Shakespeare is post-Medieval (mediaeval).

The renaissance late 1500s and early 1600s were a time of experimentation with the language, by lexicographers, writers and playwrights, so presumably by the public as well. One of the words added to English at this time (I wish I had a swear word) was "niche," for a cut-out place in a wall into which a statue, etc., could be inserted. In 1611, Randall Cotgrave put together a dictionary of English and French, with the first English use of "niche." He used the pronunciation "nitch" because the Italian "nicchia" fit his meaning better than the French "niche" which had more the sense of a grotto. His friend Ben Jonson, the playwright, used the word in one of his plays the following year but as "niece." Others quickly followed, using "nice" and "nitch" but most following Cotgrave's spelling. Now, it also means a person's place in a hierarchy, school of art, etc. (I had to research the word for use in a book (picky editor).
In Canada, the pronunciation "neesh" is popular because of the French in eastern Canada, although the word and the meaning came from English rather than French usage.
A long digression, for which I apologise, but to atone:

Byrlady (birlady) appears in a play of 1570, and was also used by Shakespeare in Romeo and Juliet in 1592 (publication dates) and it does mean "By Our Lady." There are several other spellings.