Complex sentence: Ned attempted to rescue Lyanna, but he found her in a bed, dying in a pool of her own blood after giving birth to Jon Snow that she made Ned promise to keep him as his bastard.
Split into simpler sentences: Ned attempted to rescue Lyanna, but he found her in a bed, dying in a pool of her own blood. In the show, it is revealed to be after giving birth to Jon Snow, and that she made Ned promise to keep him as his bastard.