One plot idea could be kagome getting involved in the vampire - werewolf conflict in the Twilight world. Maybe she has some unique powers from her own world that could change the balance. Another idea is that kagome and Bella become friends and kagome tries to help Bella deal with the supernatural chaos around her.
Some fanfictions might explore the idea of kagome being a long - lost relative of one of the Twilight characters. For example, she could be related to a vampire or a werewolf. This would create a whole new set of relationships and storylines. Another plot could involve kagome using her knowledge of magic and the feudal era from her own world to teach the Twilight characters new ways of fighting or protecting themselves. It could also be interesting if kagome's presence in the Twilight world starts to change the rules of that world, like how vampires and werewolves interact with each other.