Another way of saying the likes of (whomever):
like noun (2)
Definition of like (Entry 5 of 9)
1a : one that is similar : counterpart, equal have … never seen the like before— Sir Winston Churchill
b : kind sense 1a —usually used with a preceding possessive put him and his like to some job— J. R. R. Tolkien
I am, of course, quite conversant with the expressions "the likes of him," "the likes of them," or "the likes of me."
"To my like" sounds just... off. I genuinely thought it might be a typo for "ilk."
But he's the world-famous playwright, so who am I to argue, right?
Regards,