Can "what kind" be plural?

Is kind of patents ever correct?

It’s a bit informal. Kind of (plural noun) is surprisingly uncommon in formal writing, fairly uncommon even in journalism, but common in speech and fiction:

She had the kind of eyes that followed you around the room. I’d thought that happened only with paintings… (fiction)

But I don't think it faces the kind of problems, say, a Los Angeles does. (spoken)

Those kind of games tend to be tiebreakers with teams that are alike. (spoken)

Does kind of patents take a singular or plural verb?

Plural, almost always. This is like how a lot of people is plural. No one says A lot of people is upset about it. So it’s what kind of patents are, not is.

Is this the kind of thing where whatever option I choose, it’s going to sound wrong to someone?

Yes.

Then is it best to just recast the sentence entirely?

That’s up to you. I probably would, unless the context was informal.

Pam Peters examines this topic in some detail in ‘The Cambridge Guide to English Usage’. She considers the following two sentences:

These kinds of problem are to be avoided.

These kind of problems are to be avoided.

The first, she writes, ‘entails an abstract / noncountable use of the following noun (“problem”), and helps to synthesize the discussion in argumetative and persuasive writing.’ Of the second, she writes, that it ‘is simply a more relaxed form of the full plural construction, and tends to appear in interactive writing and live speech’. If, then, your example appears, as seems likely, in a formal context, then ‘what kinds of patent are being issued’ would probably be the one to go for.

The problem is the word "patents." What you want is:

What kind of patent is being issued...

It is singular because you are talking about the kind in the singular. You could pluralize as:

What kinds of patent are being issued...

So the problem is that you pluralized patents; that is that part that sounds strange.

UPDATE: Based on Barrie's answer, I gave this a little more thought. I thought a little more analysis might be useful. The phrase "kind of patent" is a genitive structure, where "kind" is the noun, and "patent" is the restriction. To be clear, the basic purpose of the genitive is to place a restriction on the word to which it is applied. The word "patent" restricts "kind". There are many "kinds" in the world, but the only "kind" we are interested in are ones that pertain to patents.

Since "patent" is not the subject of the sentence, merely a qualifier on the actual subject, then it does not affect the verb conjugation for number. However, for the same reason, it is probably the case that it can be either plural or singular. It doesn't really matter in a sense. In one case we are treating it as an uncountable, an abstract, and in the other as a countable, a concrete. A person who writes patents could be called a "Patent lawyer", but that doesn't mean he only deals with a single patent. I suppose he could also be called a "patents lawyer", but that is not the idiom. I think the same applies in this case.

However, in this specific case, because of the proximity of the genitive to the verb, it makes for a little confusion, and so, unless there is a good reason to the contrary, a singular seems the optimum choice here.

To me, in the second case "what kinds of patent(s) are..." the proximity of the singular to the plural verb is less of a clash, but I recognize that views may differ on this.

Since the patents are being issued rather than the kinds, I would say that the correct phrasing is:

What kind of patents are being issued?

Note that if you were talking to a lawyer in a patent law firm where everybody was a specialist, you could ask:

What kind of patents is your specialty?

because his specialty is a kind and not patents.

Finally, since there is probably more than one kind of patent currently being issued, possibly an even better phrasing (depending on context) would be:

What kinds of patents are being issued.