Originally Posted by Soviet
I'm 100% sure it's connected right, but the Ground is kinda just hanging there not connected to anything... Where can I connect it to? I can't connect it to a screw cause the wire isn't long enough, would I be safe extending the ground wire with another wire, then connecting it to something metal?
Yes, extend the wire by soldering or using a wire nut with another length of wire. If the ground is not connected, you may or may not get good results. The subject of grounding in antenna systems is deeper than you want to get into (and frankly far deeper than any understanding I may have), suffice it to say you almost certainly need it connected. I see four screws on the back of the unit in the mirror in your picture that would be good candidates given what we know about the system right now.
Also, in this vein, once you have a solution that works fairly well (if we can find one) you'd probably be best off to minimize the lengths of the wires in the part of the setup that you make. Meaning, everything between the coax from the Terk antenna and the Sony connector should be as short as possible for best operation. However you can just shorten them all once we finalize the connections and stuff.
Would these 2 items be all I need to solder properly?
Yes, those should suffice. Look up some tutorials online on how to solder, and maybe try it on some other wires first. Keep a wet sponge handy to clean the tip, and I like to work over cardboard so that I can press wires together against it or something and not burn my table. It's a nice convenient portable or improvised work surface. Oh, and if the iron tip screws in, don't make the mistake I did when I was about ten years old and try to tighten it with the iron still on. I got blisters on the pads of my thumb, index and middle fingers but hey, I still have my fingerprints so I guess I had good enough reflexes to let go before permanent damage set in!
One reason I suggested wire nuts (and binder clips) is that it's easier to undo than solder joints. And since you have an unusual configuration, you may need to try several things to get it to work right (not that there's any guarantee any of them will work right at all anyway to tell you the truth, but hey it's worth a shot). If you don't consider wire nuts to be an acceptable long term solution, then you can just solder connections after you find one that works using something more temporary.
Hopefully you understood that I was suggesting you try two different configurations in turn, if not I can restate it to be more clear (I'm just unsure since you didn't address that in your reply). In actual fact, I expect the second configuration to work better. That's because the signal your unit would have seen from the original antenna is unbalanced, and the balun output is balanced (a balun is called that because it converts from balanced to unbalanced, in either direction). The output from the coax is unbalanced so it may better match what the radio expects. We just need a good way to connect the coax to the wires... and you can do that by using the F-jack that I linked or else you could just connect to the center wire with an insulated alligator clip (insulated so it doesn't short to the outer ground part of the connector) and another clip to the outside part of the connector. If that setup works, though, you'd definitely want to get the jack for the permanent setup.
I'd suggest that you test with the amplifier in the Terk off as well as on. You could also play with the amplifier strength if it has a dial, but I'd probably just turn it up to half or three-fourths and test it on vs. off to simplify things.
Good luck, let us know how it goes.