A student of mine has asked, "Why are lions called 'King of the
Jungle' when they don't even live in a jungle?" I've searched the web
and have not come up with an answer.

Maybe it's a metaphorical use of 'jungle' as opposed to 'civilisation'.

But can he be 'the King' when he doesn't wear a crown?

I have always assumed that "the jungle" is used here as a synonym for "the wild."

Why "king?" I speculate that it's because the lion has no natural
predators, and is at the top of the food chain.

as was stated before, Jungle seems to refer to the wild environment.

King is not only because he is at the top of the food chain, but also
because his mane is reminiscent of a crown, and gives him an air of
royalty.