My inclination is to repair, not replace. Plastic can be tough to glue depending on both the plastic and the chosen adhesive. If I remember this part is a softer plastic, probably a polyethylene. It lends itself to easy casting/molding, is fairly durable in cold weather but is probably the most difficult to glue. There isn't a solvent I am aware of would really do any kind of job on it.
I just went down and looked at one of mine and I think the above is correct. You could try a small drop of plastic adhesive and see if it actually melts the plastic. If it does that is the route I'd go. A combo plumbing cement (PVC, CPVC, ABS) or JB Plastic Weld would be my first choices. If anything will work the JB will. Mix according to directions, just a drop, align the parts and hold/clamp 'em together until they're good and stuck. Don't pull them apart to check, it will either work or not. Let it cure a good 24 - 48 hours before using.
OK, never mind. Just read the reel description: "made of nearly indestructible nylon". This is tough but the JB may still work. I've also had some success with gentle heating until just "melty" and pushing together, a welding of sorts if you will. the trick is to keep things aligned until the nylon sets. Try to not capture any air bubbles in the joint or it will be weakened and likely break again.
A more permanent solution would be to get a machine screw that fits the nut and is about the same length as the broken part.Think aluminum or better yet brass, easy to work and corrosion resistant. Cut the stub off the mounting base and drill it out to accept the screw.
I'd make the hole just large enough where you have to turn/thread the screw through the hole, not where it slips in. this will help keep it from turning when you adjust your spool. You would likely have to do some filing/grinding on the screw head to get it to fit under the mount well and not spin. Option #2: Your rod handle looks like fairly hard thermoplastic (that means it was formed with heat). Once the hole is drilled in the mount put it on your rod handle and mark exactly where the screw head will be. Remove the mount, heat the head of the screw (not while it's still in the mount) enough to melt the rod plastic and press it into the spot you marked. Don't burn your fingers, use a pliers or Vise-Grips and don't damage the threads. Wait until everything cools down and gently wiggle the screw free of the handle. Now screw it thru the mount until it's just tight, place the repaired mount on the rod handle aligning the screw head with the recess. You may have to turn the screw a little to get it to push in. Be careful you don't upset the tight fitting nature of that recess. It will not only contain the head without having to file/grind it down so your reel mounts fits like it should but keep the screw from turning if you adjust the reel.
I know, this all sounds like a bunch of messing around for a $4 reel. Thing is your reel is worth $0 right now. You have nothing to lose and an even better reel to gain if you embrace the screw solution.
Last two things: 1.) Welcome to the Shanty! Lots of great folks and info here. 2.) If you add your location to your profile it helps us help you. Maybe someone knows a spot just down the road where you can get what you seek. Or is willing to share a great fishing spot. Or, well, use your imagination.
