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.