How long ago did you buy it - are we talking 2008 or more recently?
If it really was 2008, then yeah, the warranty is absolutely gone. Nintendo's standard warranty on electronics is typically one year from purchase, and even if you somehow had an extended plan through GameStop, those rarely go past 2-3 years. Since you're well past that window, Nintendo won't touch it. The fact that it's a known design flaw with the hinge doesn't really matter from a warranty standpoint once that much time has passed.
Your best shot would be contacting GameStop directly with your receipt to see if they have any goodwill policy for defective refurbished units, but honestly that's a long shot after 18 years. They might offer store credit or a discount on a replacement, but there's no guarantee. If you're attached to the DS Lite, your realistic options are either paying someone to repair the hinge (there are independent shops that do this) or accepting it's time to move on. The DS library is still playable on other hardware if you want to keep gaming on those old cartridges.