Connect broken headphone+mic wires
You will generally need to use a continuity tester (most convenient is to use an off-the-shelf multimeter in diode beep mode; but you could also use a battery and a buzzer) to find out where the leads go on your headset and on the jacks. There is no standard by which I could say "connect this colour to this colour and you'll be fine".

Except the blank copper strands, they should definitely all go to the ground or the sleeve part of the jack.

If we were to go without a multimeter, you need to determine whether (a) the thin wire on the left is insulated, and (b) whether the thick one on the left has a copper shield, then an insulator (can be clear lacquer) and another wire inside it. In the (a) case, that wire will be for the microphone. In the (b) case, the inner of THAT wire will be for the microphone. You can find whether a wire is insulated by lacquer by using a cigarette lighter, the lacquer will burn. By elimination, you might find which wires should go to the speakers.

Speaker part has 3 wires: left signal, right signal and common return (ground).
Microphone part has 2 wires: bias+signal and return (ground).
Return (ground) is sometimes combined into one from both parts, so some cables have 4 wires, some 5 wires, some cables have simple cables for speaker with a ground hanging loose outside but a separately shielded microphone wire.
Signal wires are always insulated; ground wires sometimes are but usually are not.

Microphone wire can have a thinner conductor than other wires without ill effect but for it to work optimally, it needs to be better shielded than other wires - sometimes you can see that in cable design, but sometimes all of this wisdom gets thrown out of the window.

Sorry if it seems like information overload, but given the amount of guesswork that will go into correctly identifying the wires without continuity tester, you will probably need it.

On the right, you need to determine which wires go to the speakers. You connect a 1.5V or weaker battery to the ground wire and the wire of which you think it might be a speaker wire, and listen for a pop in the corresponding speaker when you make the connection.

Using a multimeter is a very good idea; you can measure resistance between wires. You can get nominal impedance of the speakers from the headphone data sheet, but usually it's between 16 and 45 Ohm, and the DC resistance will be close to that, maybe up to 30% lower. If you measure two speakers together, you will see double that resistance. If you measure a microphone against its ground, you will see about 1-2.5 kOhm.