You should be careful looking further afield.
Studio Ghibli isn't really indicative of most anime in that it is mostly family friendly, whimsical and are feature films, whereas the bulk of anime releases are tv series and I'd say over 70% are only suitable for older teens and upwards.
Even seemingly innocent anime may have sexualised fanservice like ridiculously big boobed women and girl characters or panty shots.
The nearest anime series to studio Ghibli I can think of is Natsume's book of friends.
I've watched the first 3 series, they are charming but some episodes can be a bit creepy.
An age appropriate anime series I would recommend which has no dodgy stuff is K-on!
It's about some high school girls who form a rock band. The soundtrack is great too. You can probably find it dubbed as it is really popular, but I always watch Japanese with subtitles so I don't know for sure.
Fruitsbaskets has got a bit of romance in it. Your Lie in April mentioned upthread is a romantic tearjerker. Tribe Cool Crew, about street dance. Carole and Tuesday is good about 2 female musicians, good soundtrack again. Laid Back Camp and Silver Spoon are good. Little Witch Academia is suitable too.
For adventure anime there is Hunter x Hunter, Full Metal Alchemist etc, but this genre can get darker through the episodes.
Asobi Asobase is a good comedy for gross out humour.
Good sports animes are Crossgame, Prince of tennis, Slamdunk, Kaleidostar. Sports animes are normally very moral aswell; about overcoming adversity for example.
There are many new anime series and sequels released every season. Its a huge area of entertainment.
Hope you find something for your daughter to enjoy. You could find some series for yourself aswell!