Particularly based on the selection of entries in the New Oxford American Dictionary.
Approximately as many words as there are in English, or more. If we had a complete list of English words so organized, we just count them up: the problem is that nobody has integrated and differentiated the English lexicon in that fashion. Dictionaries (esp. good ones) tend to maximally differentiate so that you get separate sub-entries for "cow" as a verb versus a noun. The Oxford English Dictionary tends to over-differentiate, though not unreasonably, when words are a bit vague.
I think there are three core concepts expressed by "cover". One is about putting a lid or similar object on something, one is about taking responsibility or "accounting for", and the third is a term of art referring to a derivative work in popular music. There is a real-world difference in what things refer to in the case of "cover (lid)" as a verb, versus a noun, and within the realm of nouns, it can specifically be the word for "hat" in a military context (as well as many other physical objects used to cover something). One theory of concepts is "maximalist" in the sense that it would include all of the notions of "cover as lid" under a single concept, rather than further split it into "the act of..." versus "thing that you use to....". If we say that there is just one word "cover", we also have to say that it has at least three senses. We can try to reduce these uses to a single meaning and derive the other uses from a rule, but I don't see any non ad hoc way to derive the "take care of, subsume" meaning ("This is covered in chapter 2") from "put a sheet on" or vice versa.
There are a number of different words that refer to the same piece of furniture known as davenport, sofa, divan, couch and settee which as far as I can tell refer to exactly the same thing. There are also words like "love seat" or "futon" which refer to similar things, but not exactly the same thing. Using the "what does it refer to?" test, as I understand the objects, a "love seat" is a specific kind of small couch only good for 2, and a futon has specific compositional requirements, so these are different concepts. The number of referential synonyms in English is fairly low, even if you merge metaphorical and more literal uses (e.g. "pig" but excluding "pig iron" which contemporarily has nothing to do with swine, but apparently originally was connected to swine in Middle English)