Does it have to be 13+? The content rating is the minimum age, so if your daughter is 13, then anything rated 13 or lower is appropriate for her.
Both of the dramas that you mentioned are about high school students, and both have some romance, so based on that, she might like Someday or One Day or Hana Yori Dango.
I'm not sure why Hana Yori Dango is rated G on MDL, but it's about high school students, and it's one of the most popular J-dramas.