No, you can't just simply leave out the "which is" from the "which is why", because these two have different meanings. "Which is why" = something happened, because..., as a single "why" is used for questions.
The sentence
"My television was broken, why I spent the evening reading a book."
is wrong. The second half of this sentence would imply a question, i.e.:
"He asked why I spent the evening reading a book."
But as you said, you could use the "so" in this sentence and it would be correct:
"My television was broken, so I spent the evening reading a book."