the impossible badge requires an enormous amount of luck. The strategy that got me the furthest is buying only shields and cantrips and ignoring the splash upgrades. I try to enter the gauntlet with at least 5 cantrips and shields. with that i usualy get to level 13 or 14 if i'm lucky. but level 14 just has way to much blight so i can't beat that level ;(.
 "Proke"){kind=icon}
 "Pawnbarian"){kind=icon}
 "Mighty Knight"){kind=icon}
 "Christmas cat"){kind=icon}
 "Bubble Tanks 2"){kind=icon}