Shiro no Monogatari

Yasha is a very brutal character. One day he steals some relic and triggers space distortion resulting in the human world.

The heroine was dying of a heart disease at this time the the jewel that Yasha dropped healed her completely. He tries to take the jewel back, but the girl gets to die in that case. She manages to persuade him to wait for a week and live together for that time. However, it turns out that jewel is not just a precious item, but a "jewel of bliss" that can fulfill one wish.