I looked it up and most forums link to http://semarch.linguistics.fas.nyu.edu/barker/Syllables/index.txt, an NYU site that no longer works. I would like to know how many unique syllables are used in the English dictionary (not possible syllables, but actually used syllables).
To answer you question, according to that paper on that URL: How many unique phonemes single syllables are used in the English language? 15,831
I've visited that URL 5yrs ago in a wild search of a syllabary of the English language.
Anyways, you can view the paper still using the WayBackMachine one Archive.org. It went offline somewhere between Sept 23 - Oct 8th 2016.
Here's a link to the last snapshot.
The main question is only really answerable for a specific dialect and accent of English. "American English" has a lot of regional variations, and also variations by register.
The basic reason for this is that English is a pluricentric language, and even within a supposedly standardised version, it is very much permissible to borrow words from other languages and coin new words (often from roots in other languages). These new words often have pronunciations that are mangled versions of the original word or root, so the inventory of syllables is subject to random enlargement by users of the language.
English dictionaries are descriptive, not prescriptive.