I got started on the letter, but I haven't finished writing it yet.
The ending かける indicates that an action has been started but has not been brought to a finish or an end. However in this example I am puzzled by the second half of the sentence.
If I break it down literally, this is the way I comprehend it:
手紙を書きかけたんですが、I started (but did not finish the letter), but...
まだ書いていません。 still am not writing (the letter).
Assuming the highlighted translation is correct, what am I misreading in the second half of the sentence? I am puzzled by the use of the negative in the second half, as I would have assumed まだ書いています would make more sense to correspond with the desired meaning of the translation. I guess in this example using 書いています would have expressed a completed action, so 書いていません means "not finished writing" rather than "not writing"?
Let me try making another example:
I started to make food, but I still haven't finished yet.
Compare the latter example to this:
A Did you make food?
B Not yet.
(perhaps implying that food preparation has not yet started?)
If one writes the original sentence in the positive how does the meaning change? Would the following translation be correct?
I got started on the letter, but I'm still writing it.