Honestly, the stuff thats still frozen in the middle should be fine. Was your baby full term? If so then you should be ok to refreeze. Maybe mark the stuff that has been refrozen so you know, if there is a problem.
The Womanly Art talks about the realistic use of breastmilk. Refreezing partially thawed breastmilk was one of the topics covered. "As long as it has a frozen core, its ok to refreeze it" p308
Bottom line is you know your baby better. If your little one gets sick from everything, then maybe you should dump it. If not you're probably fine!