If the computer recognizes it the cable is likely good. Not charging is frequently caused by the charger plug. You should probably replace the plug, if you cannot fix it easily. Can you return it?
An alternative, e.g. if past replacement date, is to use a Dynex DX-UA Universal charger with your Asus cable. They are $30 at stores like BestBuy or $8 or less on ebay. This includes a female USB plug for your Asus cable and provides 11v, which is slower than using the original adapter 15v, but it works.
So far, Asus is not selling replacement cable/charger kits. These kits are listed at $40, which is too high in my opinion.
Good luck.