... the stainless wire won't go more than about half way down.
Is the end of the wire blunt? If so, then filing the end to a chisel shape, and using it like a bradawl, might be more effective. Alternatively, if the narrow bore is only at the nozzle end of the iron's tip then, with the tip removed, it may be easier to clean from the other end (perhaps, using a fine drill bit twisted between your fingers).