Sure, buy a $10 iron and solder that puppy back on.
Clean up the end if the wire first, cut the wiring back to eliminate the frayed end, twist it together nice and neat, and trim back the insulation just a wee bit, say 1/4 inch of exposed wire. then, to make the job easier, heat the end of the wire with the iron and then apply a drop of solder so that it coats the wire. This is called 'tinning' and makes re-attaching easier.
Don't have to remove the old solder, the iron will re-melt it and it'll help reattach the new wire. You'll need very little new solder. Of course if you're really anal you could use solder braid or a solder sucker to melt and remove the old solder, remove the bit of broken wire, and resolder the wire in place.
Finally, whenever you're applying solder, don't touch the solder directly to the iron, heat the parts and then apply the solder to the parts.
mystic_1