There actually are synthetic fibers that are heat resistant and that you can use things like flat irons on (just did it myself the other day). Do you happen to know what material your wig is made of?
Finished: Armin Arlert, Mayoi Hachikuji, Tomoko Kuroki, Ultimate Madoka, fem!Masky
In progress: Kougyoku Ren, Lightning Farron
Upcoming cons: Sabakon, Youmacon, Con+Alt+Delete