Either the battery is defective or the charging light is simply broken.  Will the system turn on or charge the battery at all?

You will probably need to get a repair.  However, if you have not downloaded any games or added money to the eShop, you can simply try to get an exchange at the store where you bought it.  If you have spent money on the eShop, then only Nintendo can make sure that you keep those games.  You can schedule a repair by calling Nintendo at 1800-255-3700 (6am-7pm, Pacific Time, 7 days a week).  The repair will be free and typically takes 7-10 business days.

Warranties are usually activated when a store clerk scans the system's barcode, but since this was sold by an online retailer, it is worth asking them exactly how they activated Nintendo's warranty.

I buy most of my games online, but for big ticket items like consoles I always buy local, like at GameStop -- just in case -- since most local stores offer at least a 30-day warranty for free, which covers defects such as what you're experiencing now.  Sure, it may cost you $12 more in sales tax, but it's worth the peace of mind.

Hopefully you can just send it back to that online retailer for a refund, then go to a local store and buy a new one.