Well no of course you won't be able to use your US bought iPhone 5 in China. A contract free iPhone is NOT the same as an unlocked iPhone and in the US the officially unlocked iPhone 5 is not available right now.
You need to contact Sprint in the US and ask them if they can unlock the phone for you. There is absolutely nothing else that you can do, as locking of phones is the carriers responsibility and nobody else, so Sprint are the only ones who can help you. Most phone companies require you to have a current contract with them before they will unlock your phone for you and this might be the case with Sprint. You will have to contact them - they are the only ones who can help you with this.
Unfortunately, as you have already found to your cost, there is no "global" phone available for purchase in the US right now.
Bottom line - you need to contact Sprint, otherwise there is nothing that you can do. You should not really have bought the phone in the US. If you wanted to use the phone in China, then you should have bought it in China. Why did you not buy the phone in Hong Kong - they are available there?