After reading some posts on ScienceMadness.org, I determined which steps would need to be taken to purify the manganese dioxide. I first washed the manganese-containing battery paste with water and vinegar, then dissolved everything in HCl. I filtered off my now-MnCl2 solution from the carbon, but it was contaminated with iron. NurdRage's selective precipitation procedure came in handy for resolving that issue, and finally, I got a pretty pink solution - pure MnCl2!
Even better, I recovered some very beautiful shiny lumps of manganese metal. While the thermite only gave a 23% yield on account of being so violent, it added another element to my collection, which is something to celebrate for sure.