My recommendation is to find a replacement charger.
A quick check on eBay shows that model is popular and varies in price.
At this point you need some way to verify the AC charger. A replacement charger, used, is about $15 to $30, which is a bit cheaper than a new battery. So buying a new charger would be the cheapest way to verify the charger, unless you have a volt meter and know how to test the charger.
A used Asus X551C Notebook goes for about $ 200 on eBay,. But you would have no guarantee. But one vendor claims he will sell a warranty for one year for $35.
Another test would be to find some way to charge the battery without using the charger. If that resolves the issue, then you know the charger system is not working right.
Part of the charger circuity is inside the laptop. So you could have a good charger, and good battery and still not be happy.