Yes it came with the kit you just have to wik it yourself I think I might give it a shot
Can't hurt to try, it will take a bit of practice, so if it doesn't wick well/work like you'd like the first time, don't get discouraged.
Wire and wick is dirt cheap (~like $6 for about a 5 year supply of wire, and ~$12 bucks for several years worth of cotton).
If you decide to attempt to wrap a new coil and don't have access to a rod/jig, a piece of regular spaghetti (LOL) can be used, in a pinch, to wrap an ~2mm coil.
It will take some trial an error to figure out the best coil/wick configuration but once you figure it out, it should be easily repeatable on that unit.