Sorry if this is just an obvious thing that everyone already knows.
I was thinking the other day. It was actually pretty LUCKY that Anna was turned to ice at the exact time she did. Click this link and watch it. You'll notice that at the very moment Hans' sword collides with her fingers, she turns to ice, thus shattering the sword and creating a blast that knocks him back. However, if she hadn't frozen then, the sword would have simply sliced through her hand. Having already been weakened in the first place, Anna would not be to do anything other than cower next to Elsa, and most likely trying in vain to staunch the constant flow of blood. And THEN she would freeze.
Hans would just ignore this (perhaps let out a frustrated huff) and attempt to kill Elsa a second time. Although, as I'm writing this, I realise that, once Elsa noticed that she was being MURDERED, she would defend herself.
A number of things could happen next. I suppose that Elsa could be overwhelmed by grief or angst and simply not care anymore. Or she could be overwhelmed by grief and anger and turn into a berserker to destroy Hans. One thing's for sure though. The movie would be at least a couple minutes longer if Elsa had to defend herself.
But all things considered, the chances are that it's better that Anna froze when she did. To be honest, if her hand got sliced like that, it would not be much of a kid's movie anymore. And let's face it, she was going to be saved anyway.