Bunu bilmedim or Bunu bilmiyordum?

This is one of the common mistakes that people do while learning Turkish Language. When you did not know something, you do not use the past simple tense in Turkish-which is I did not know this-Bunu bilmedim. 

We do not say "Bunu bilmedim". But we use the past continuous instead and say; 

"Bunu bilmiyordum"

Bunu biliyor muydunuz? Did you know this?

Evet, biliyordum. Yes, I knew.

Hayır, bilmiyordum. No, I didn't.

So, when you just want to say I knew it/I did not know it, you should always use the continuous form.

But, can't we say ever 'bildim' in Turkish?
Yes, we can.
Let's say you are in the class and the teacher asked a question, and you solved it.

Sorunun cevabını doğru bildim.
Either you say 'doğru' or not, this sentence means 'I found the right answer of the question' .

İlk önce ben bildim. I was the first who told the right answer.

So, it is about something that happened for the specified time at once, not like biliyordum which covers all the past.