I thought it was obvious that the reason MJJ was arrested for murder was because he insisted his own guilt. Even helping disguise the crime scene. A confession along with incriminatingly being at the crime scene is often enough to convict someone. Not to mention this was like during the 90's.
Poor boy was so in love with that girl and she didn't want to be remembered/forgotten as the "loser girl who eventually died in a loser manner and killed herself because she was a lame loser". She had issues like severe insecurity, poor self esteem, identity crisis and overwhelming social anxiety. So she wanted to be remembered with impact and have her death be declared as a murder. It helped that there had been an actual homicidal stalker after her life.
Also, I think the cops could've easily been called beforehand by one of the characters - (they did all suspect there to be an actual murder).

AnewGuy:
Came here looking for that answer too. I guess they just forgot to adress that. Also as soon as she jumps to her death the cops are nearby and ready to arrest Mo Jun Jie. No evidence to convict him. I liked the drama but there's just a few things they could have fleshed out to make it a bit better.