Online services have been permanently discontinued on Wii and DS.
You can thank the disbanding of the company, Gamespy, which hosted Nintendo's online servers.
Today, for 3DS and Wii U, Nintendo has their own online service, run by themselves.
But they aren't going to expand that to cover the Wii and DS again. If not for technical reasons, then for monetary. The Wii and DS are no longer in production and Nintendo makes no money off of them anymore, so there is no incentive for them to start up the online servers again.
Consoles have a lifespan of 7-10 years, and the Wii and DS had their time.