How to replace woeful radio antenna with proper one?

We have a cheap-'n-cheerful home stereo system (model E238.1.A) from Selective Marketplace, that was (& still is) offered all over the place in newspapers & little catalogues, (curvy silver plastic design).

The radio will only get the few national stations with the strongest signals, and I think the antenna's the culprit. It consists of a very thin, brittle wire that breaks easily, inside a very thin plastic coating. I've had the back of the unit off (guarantee expired), and this wire continues (with a knot to keep it in place) to the metal framework inside, to which it's screwed down.

We have a proper exterior aerial from a previous radio, which got very good reception and is ready to re-use. I tried clamping its coax cable to the original antenna wire, and there was some improvement. There was also some improvement when I tried replacing the wire internally with co-ax simply touching the metal frame in the same place, with a jack on the end to connect the aerial. But I think it was just because there was now some shielding on the cable, because I don't know how to connect the coax to the works inside.

Could anyone tell me where to connect the coax core and sleeving wire internally?
I could do a photo(s) of the innards if it helps, but I was wondering if there's some general method for all radios?