I think this site may be help you it's illustrated stories with voice the narrator.

http://study.jr.naver.com/donghwa/index.nhn?ageType=ALL

You are probably more advanced than I am.  I started learning Korean 6 months ago.  However, I have been documenting everything that I find in my language journey on my blog. http://jreidy17.wordpress.com/

I have learned a lot from children's resources, such as "Teach Me Korean" song book and Tuttle Korean Flashcards for Kids.  

My Korean pen pal showed me the Educational Broadcast Systems free website for learning Korean which contains videos to teach both children and foreign-born wives

http://jreidy17.wordpress.com/2014/06/03/ebs-offers-free-korean-language-learning/

The tricky part is signing up in Korean, but just ask a Korean to help you if you get stuck.  

Today I was listening to a funny American Whitney who has lived in Korea for 3 years.  She had just the right balance of a good accent but not too fast speech, so I could understand her Korean.  See "Youtube, not studying" post.  I do listen to native Korean speakers, but some of it is so fast!

Hope this helps.  

When you decide to buy a book, it's also important to check whether the author of the book is reliable and to see the index. Books with titles such as Mastering Korean within ~days or months are nothing but hype. It takes a lot of efforts and dedication to reach some point at a language.

Also, when you get a book, it's better to read it all the way through first rather than buying another book of the same category before you finish reading it.

For example, when you are reading a reading comprehension book, you can underline parts that you find difficult and then refer to a dictionary to broaden the vocabulary or usage of particles(marker) or ending. If you are reading a grammar book, you can ask more questions about the grammar phrases on italki or other sites.

I think books from Korean Language Institute Yonsei University are reliable materials. Those books have in-depth explanations and cover wide aspects of Korean.

Back to your question, my recommendation among kids' TV programmes is 뽀로로. Voice actor and actresses talk very slowly and they don't speak too difficult vocabulary or childish babble. http://www.youtube.com/watch?v=EgNK-1n4XoY

Korean is a very context dependent language.  The language spoken to kids is different in vocabulary, tone, and structure to conversations among adults or peers.  Also, there is a polite form of language for strangers and to show respect towards family members, leaders, and the elderly that will not be easy to notice in children's materials.  Unless you are learning Korean only to communicate with children, I suggest checking out the free materials provided by talktomeinkorean.com

Hey Thaddeus, thanks for your comment too.  I agree the language will be different, but as I just explained to Shao, my problem at the moment is picking out words in a sentence.  For example, yesterday my Korean tutor on here spoke to me in Korean.  When she over what she said word by word, I knew most of the words, but I just couldn't pick them out in a flowing sentence (and she was speaking clearly).

I've bought loads of stuff from talktomeinkorean.com (and many other books and online videos and Rosetta Stone, etc), and they're all useful in their own way (and in my journey to learning Korean, at different times), but there's a big difference between two tutors having a practice dialog where they've just told you what every word means to just hearing normal Korean where I know I won't know everyword but at the moment I even have difficulty picking out the words I do know.

Does that make sense?  I asked about kids TV because I think they'll likely go slower while talking correctly (I'm thinking like Peppa Pig for English - for example https://www.youtube.com/watch?v=Il1RX6_h9Xc)