Hi J.,
I agree with the water your doctor recommended. A little juice in it is fine...anything to get her to drink it. Make sure she has a sippy cup of water available at all times. I have noticed that my little 14 month old will down the water, it I constantly offer it to her. However, she doesn't ask for it if it is not offered.
I would also try to get her to eat foods that naturally soften the stool. You might want to give her some rasins or prunes (or prune juice). I know that sounds gross, but she doesn't know.
Keep her away from cheese for a while, as that is known to "stop you up".
And finally, if all else fails, try a little bit of mineral oil. Put a teaspoon of it in her morning juice. My little brother had that problem (he would actually hold it so long that when it finally came out, it was too big). His pediatrician recommended that my mom put a teaspoon of mineral oil in him morning juice. It worked wonders. However, this should only be used after you try a diet change, like mentioned above.
Good luck!
T.