I guess it depends on if you want her to watch a lot of romances at her age (I'm sure she's into that now) or if you want her have more friendship, mystery, action adventure or fantasy and/or innocent romance type shows. If you are thinking romances I would go mostly with strong female leads for sure as you don't want her to have that wimpy girl in love syndrome (as in "I'll put up with abuse and do anything it takes for my man to like me", type of crap). There are always exception to the hot and spicy rule (cause I like em like that - you're beautiful and Yamato Nadeshiko Shichi Henge are examples of hot and spicy guys that don't annoy me because you get to see a lot more from the guys prospective in these and know what their thinking where as in shows like goong and playful kiss where almost all is from the girls prospective and you rarely know what the guys are thinking and they just come across as butts. And yes, I have several listed below that fall under the hot and spicy category, I couldn't help it. Here's my list, I think most of it's pretty decent stuff.
Teenish with little or no romance
Baby and me
Death note(based on anime)
Death note the last name(based on anime)
L: change the world (death note) (based on anime)
Bloody Monday
Ghost Friends
Punch
Dream High (I haven’t seen but heard it was good)
Teenish with mostly innocent romance
lovely complex (based on anime/manga) Honestly I didn't really like this as a live action but the anime was pretty good
Skip beat(based on anime/manga)
Ouran High School Host Club(based on anime)
Mei-chan no Shitsuji (based on anime/manga)
Marmalad Boy(based on anime/manga)
Ikemen Desu Ne or You're Beautiful
Hana Kimi (not sure about anime but for sure based on a manga)
Buzzer Beat(based on anime/manga)
K-POP-the ultimate Audition (only watch a few episodes but the girl is a strong lead)
Yamato Nadeshiko Shichi Henge (based on anime)
Hana Yori Dango (based on anime/manga)
Older but fairly safe romance
Sungkyunkwan Scandal
Tamra, The Island
HeartStrings
Rooftop Prince(only seen half of them so far)
Usagi Drop (based on anime, not so much on the romance but it was cute)
Smiling Pasta
Nodame Cantabile (haven’t watched but heard it was ok) (based on anime)
Protect the Boss
There are more but this is just what I came up with that a 13 year old might like. I watch mostly romances myself so I don't really know too many appropriate ones that have action and adventure stuff but I know they have them. Also when I was that age I watch a lot of mystery shows and there are a lot of those in the Asian section also, even some that are kinda teenish.
Hope that helps.