Changing microphone's jack to headphone's jack
There are usually 4 wires with headphones (those with microphones included) - usually red (right) green (left) black (ground) and white (mic). You want to double check this with your specific model, shown further down. In essence you want to leave your headphones mic disconnected and rather connect the white (if it is mic) to one of the 2 microphone wires - the positive one. If you know which mic wire is ground, then connect that to your headphone jacks ground too - the remaining one should be positive.

Conventionally the 4 pole jack is as follows - imagine it pointing up as in the diagram:

 ^
| | Tip - left ear positive
---
| | Ring - right ear positive
---
| | Ring - Ground - shared ground for right/left/mic
---
| | Sleeve - Microphone positive
| |
___ Sleeve insulation


I suggest you put and hold a small blob of solder on the 4 wires, just enough to burn through the insulation and tin it. This isn't supposed to cut the wire, just create a conductive bit that you can confirm which wire is connected to which pin on the headphone plug - use continuity for this and create a mapping on paper to help you remember. (Word of warning - there WILL be some resistance between the two ear speaker wires and ground - since its a coil. When you check continuity, make sure the resistance is zero and not 14 or something Ohms.)

On your multimeter, check continuity between the ground and the tip, and you should hear a crackling in the right ear. Then do the same for the top right - you should get a crackling in the left ear. This way you confirm which pin is ground, and the two ears. The ear pins aren't needed for your task, but its nice to know.

Now the remaining wire is the mic - all you need to do is bridge the mic wire with  that wire on the plug side, and bridge the mic ground to the shared ground. I am assuming you haven't cut off the headphone plug, and simply stripped the insulation. By the way, if you DO connect your mic, disconnect it from the headphones mic wire so there isn't interference.