Hi Ian,
There's a whole series of things all connected together to charge the battery, so it's a matter of figuring out which of them is causing the problem when it won't charge.
Generally it's easiest to start at the charger. Make sure your wall outlet works first, then verify that the charger is putting out the correct amount of power; in this case you should see 5.2VDC on the output pins using a voltmeter. If you don't have a voltmeter you'll have to either try a different charger or try the charger on a different DS or Gameboy.
Once the power to the DS has been verified, the most common problem left is that the battery has died. Given that the last DS was shipped before 2010, it would certainly be no surprise if the battery was shot by now. Given the ease of replacing the battery, the availability and low cost, it would make sense to replace the battery as the next step. Only if that doesn't work will it become necessary to open up the DS and start checking internal parts and components.
You can always try checking the charging port for dirt or debris. It can't hurt to clean it with 90% or higher isopropyl alcohol and try pushing the plug in different directions when it's inserted into the jack to see if maybe there's a loose connection.
Give some or all of that a try and let us know what you find.