A bit surprising interpretation, but the grammar allows for it, so yeah, we added it.
In that meaning - shouldn't it be rather "I sent the beautiful postcard of hers" ?
I believe tj1983 means something like "She asked me to send a beautiful postcard written by her to her grandma, and I did send it".
Oh right, sorry br0d4, I somehow interpreted your version as "a postcard with her photo on it"...
I think we just see so much more genitive than dative, it didn't even initially occur to me that "jej"was also dative. So thanks - I'm glad I'm not crazy