Tony could also help you with the soldering tips, but I use plumbers silver solder and a propane torch.

I built a weed eater boat from scratch. This is how I got a solid shaft attached to the end of my string trimmer flex shaft so that a common r/c boat prop would mount to it.

Using the flex shaft from your weed eater:
Cut the end of your flex shaft, you will find that the shaft has an inner core that is not bonded to an outer sleeve. (This will be more noticeable as you follow this procedure) Bend the shaft, flex it side to side. The more you do this the more the outer sleeve will relax and lengthen. It will appear as if inner core is retracting inside the cable. Continue to work the cable until you develop an outer sleeve, at the end of the cable that was cut, so that you have at least a 3/4 of an inch recess. Then you will find that a 3/8 steel shaft will slide perfectly into the middle of the flex shaft (get a length of 3/8"steel rod from the hardware store for this). Solder or braise the shaft in place. I used silver solder and it held very well. I was able to obtain a prop from a r/c hobby store that requires a 3/8" drive shaft. If you are like me you may have to practice or experiment with this idea before you get it right. But broken weed eaters are a dime a dozen and getting an extra one or two from your local mower shop should be no problem.

Hope this helps.