Let's look at a concrete example before going deeper.
Consider your screen name.
Saying that you are a cookie monster conveys the idea that there is a group of entities that are each called cookie monster, and you are one of them.
Saying that you are the cookie monster conveys either that the 'group' of entities really has only one member (you), or that you are the most outstanding member of the group.
In each case, the focus is on some kind of classification scheme.
Saying that you are cookie monster says something about you personally - you really enjoy cookies, eat them messily, etc.
It's a similar case with your example. "I am king" uses a null article (not a zero article).
The zero article is the most indefinite article, and the null article is the most definite. Peter Master arranges articles in order from most indefinite to most definite:
zero (Ø1)--some--a--the--null (Ø2)
- Peter Master, "Acquisition of the Zero and Null Articles in English", Issues in Applied Linguistics, 14(1)
Here's an example of the zero article and null article from the same paper:
- Zero article: The boys ate chicken.
- Null article: Mr. Jones was appointed chairman.
The null article example has a similar quality to your "second king" example.
Note that both zero article and null article refer to something that is absent from the sentence. It can seem a little odd to describe something missing as potentially having two polar opposite possibilities. Masters goes on to say:
The zero and null articles can be readily distinguished by their paraphrasability by either an indefinite or a definite article, respectively
That is, if the sentence retains its sense when you insert an indefinite article, the original had a zero article. And if it retains its sense when you insert a (the) definite article, the original had a null article.
With your example, "... I am king" has the sense of "I am the king", with everything that being king in that context entails. But that sense is lost when "... I am king" is paraphrased as "I am a king". The original therefore has a null article.
Can anyone explain why there is no indefinite article in front of the second "king"?
That's because the sense of the second "king" is a matter of identity, requiring the most definite article - the null article.