Purification/Extraction of Sodium Permanganate
You are correct. First step produces $\ce{K2MnO4}$, then you need to oxidize it.

My first guess was that adding $\ce{H2SO4}$ will cause disproportionation of $\ce{K2MnO4}$ into $\ce{KMnO4}$ and $\ce{MnO2 + K2SO4}$. Filtering removes $\ce{MnO2}$. Crystallization will give you pure product. Or you can use a mix of 
$\ce{KMnO4 + K2SO4}$ for many purposes.

Industry uses more efficient method: electrolysis of $\ce{K2MnO4}$.