This is entirely acceptable in speech and informal writing, and possibly acceptable in all but the most pompously formal writing.
A coordinating conjunction won’t work here: the first piece, What’s even worse, is not a sentence but a “free relative clause”, a construction which almost always acts as a noun phrase.
There are three ways of understanding this sentence:
What’s even worse acts as the subject of a sentence in which the copula and the complementizer that have have been omitted.
What’s even worse [is that] I slept right through my stop.
What’s even worse is a bare NP announcing a topic, which is then defined by the subsequent clause. You could repoint it to express this reading:
What’s even worse: I slept right through my stop.
What’s even worse acts not as an NP but as a sentential adverbial, similar to Furthermore or Moreover. The same thought could be expressed by the bare adverb Worse:
Worse, I slept right through my stop.
It doesn’t really matter how you parse it. It’s just an idiomatic way of saying that what happened next was even worse than what went before.