I would sharpie that if you can. It will take a while, but it's worth it in the end.
If not you can use the spray dye method. Since it's a white wig, it should dye the correct color. I would make sure it's a wig that's good with dying first though. You could easily mask off the rest from the bottom and just dye that, then spike it after. To mask it off, you could use a nice dry clean towel, that is small but covers the whole head, and cut out the sections where the purple will be, then use painters or masking tape to take it off. If the tape gets on the hair it will not ruin it like other tapes will, but it will stop the dye from getting to the rest of the hair..
Cosplay DA Regular DA
-Fate Testarossa (Mahou Shojo Lyrical Nanoha A's) ~90%-
-Ryouko Ookami (Ookami san and her 7 companions) ~75%
-Amy Sorel (Soul Calibur) 25%-