I went through the same thing with my son who didn't walk on his own until 18 months! Starting around 13 mo, he would walk holding on to something like your daughter. We'd go for long walks holding his hand and he loved it. But he was and is and has always been such a cautious fellow! He's now 32 mo and just the other day for the first time, he climbed up the slide at the playground all by himself. I've seen babies blow past him while he was deciding if he was ready. When we took him for his 18 mo check-up, the docs wanted him to have an evaluation and send someone to help him start walking. i was pissed! I knew that he was fine and it was totally psychological. I knew that he'd walk when he was damn well ready. And let me tell you, when he did start taking those first steps, he was an expert walker almost instantly. We skipped that whole clumsy cute falling-down getting into everything stage altogether. Soon after he was running.
At least one other mom mentioned the idea of playing with other kids. That was exactly what did it for my son. The neighbors had two boys ages 9 and 6, and they loved my son and always wanted to play with him. they'd pick him up and hug him. So cute! One day while playing with them he took his first steps. As much as I loved some of the suggestions on here, and think they sound like good ideas, if your daughter is anything like my son, he is stubborn as a bull and gets performance anxiety. I might add he's also extraordinarily smart, and he knows if I'm up to something like trying to get him to walk. The first time he peed in the potty, I was in the other room! But oh... when he is with other kids, there is a whole host of things he will do that would never fly if it was just him & I.
Don't worry about your sweetie, don't get frustrated, she will be walking soon enough and then soon you won't remember what it was like when she didn't. One thing my son has taught me is that they all have their own little timelines... I sincerely think it is best to let them walk these lines at their own pace.