This was common with the original Nintendo DS models.
The screen has failed and needs replacing.
Unless you can aquire a faulty DS of the same model type to strip down and use as spares your only other option is to purchase a replacement screen, which are not cheap.
Replica screens (non-genuine foreign) are available on eBay, mostly from HongKong sellers, and despite saying Genuine part i can assure you THEY ARE NOT, but if you can get one cheap enough then by all means fit it if you are confident enough.
Otherwise, it may be cheaper to replace than repair.
Drew
DrewKrew, March 2011