Please explain why the word का is needed?

I am pretty sure का is there because का can be another word for 'cost.' 
I think the एक कमरा कितने is a typo. I think it should be एक कमरा कितना if I'm not mistaken.

Hope this helps

एक कमरा कितने का है? is a correct statement in Hindi. It is 'How much cost of a room?' or What is the cost of a room?

 'How much is a room?' in English itself is not correct/complete. 

कितना is about how much? 
कितने is how many?

 

Thank you Sudhanshu!