Ario Koyama comments

According to Asahi newspaper:

「7日朝に母親が自宅にいる中込さんを見かけた後、行方不明になっていたという。本人の携帯電話の位置情報をもとに兄の友人らが捜していたところ、12日夜、自宅から約450メートル離れた竹林で発見した。」

"After her mother saw her in the morning, May 7, they didn't know whereabouts. While based on the GPS location, her brother's friends were searching for her. Then they found her body on the evening on May 12 in a bamboo forest about 450m from her residence."

I understand that they didn't report the police of her missing for some reason, and they called the police after the very body had been found.  

That means the date police get into the incident was May 12, 5days later; the matter happened on May 7. And the police didn't open it public in the hope of early resolution considering the family's will to keep silent. But without any result, after another 7 days, the police decided to release the news on May 19 and have established unusual 50 members special investigating team by now.

Some other sources tell that she was so open-minded and capricious, like a butterfly.  

I guess the family wouldn't think the matter seriously and were reluctant to tell others of her missing because local Japanese people tend to avoid bothering others within their community.

According to the Yomiuri newspaper, the suspect says 「殺してやろうと思って刺した」 "I wanted to kill her and decided to stub her."

Because the whole thing was caught on security camera and the police arrested the man.

No they didn't. The police can't look at the security cameras in Japan. Security cameras are only used during trials.

Yes, they did. The police of Japan have the right to look at any security cameras if somebody claimed something is wrong, and if necessary, they can arrest the suspect on the spot.

Of course, the contents of the camera are used during trials as evidence.