Agreed on talking to the stablehands and the classmates to get more information, and handwriting samples to compare to the letter (get a sample of Juhino's handwriting, too. Surely some of her school notes are lying around). Also maybe the boyfriend's employer and any friends/family of his that might be able to confirm the relationship was definitely broken off? And how they perceived the boyfriend feeling about said break-up?

It might also be good to ask her mom if there's any special significance to the necklace, while they're asking about her emotional state.