offf ya ..) birşey değil.

Here it says that present perfect tense is used for actions which started at past and still going on. We don´t have such a tense and we use present continious tense instead of it. So translation must be like this:

"çalışmıyorum" or "çalışmamaktayım".

If I am wrong please explain my mistake.I am not very good at grammar. Have a nice day.

 I am not against your point totaly. But if you look at the whole sentence;

 I haven´t been studying Turkish as I have been feeling ill and down recently.

 Son zamanlarda hasta ve kendimi bezgin hissetiğimden türkçe çalışmıyordum.

you will see that in this context it should be "çalışmıyordum". Lets put your one in the sentence;

 Son zamanlarda hasta ve kendimi bezgin hissetiğimden türkçe çalışmıyorum.

I think here "çalışmıyorum" does not sound good.

 If the sentence was just like this; "Son zamanlarda türkçe çalışmıyorum." then its ok.

I agree on your grammer point but we should look at the whole sentence to make it right.