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Small amounts of blood (such as flecks or streaks) are to be expected with hard stools that include the child straining to poop. Large amounts of blood would be uncommon and would require prompt medical attention.
You may try either prune juice (excellent suggestion) or Miralax. Miralax is exceedingly safe and can be titrated to two soft stools per day. Every child is different and your daughter may need anywhere from 1/2 cap every other day to 1-2 caps twice a day.
I would encourage you to be prompt about resolving this. If left unchecked, it could become a serious problem, as well as cause behavioral problems such as developing a phobia of moving her bowels (due to the pain) thus causing fecal retention, worsening the problem, causing accidents/explosions, and starting a vicious cycle.
Christopher J. Russo MD FAAP
Director of Pediatrics
Director - Pediatric Hospitalist Program
Centra Medical Group
Centra Lynchburg General Hospital
Lynchburg, Virginia