RE: XPS 15 9560 Headphone jack function as a mic jack?
I think jphughan could be right on this (regarding headset option, or possibly the adapter not being recognized properly) and unfortunately I haven't found a lot on this yet. Do you know have a link to the adapter you bought? Also, if you just select mic by itself without the headset portion attached (so mic into adapter into system, and headphone portion unplugged) does the system at least acknowledge the mic then (assuming you selected just mic)?
One other thing you could try is removing some of the audio settings/drivers, but I would need to check into that first to see if it is worth it, so trying to see if the mic is working first might be the best place to start.
On a side note, another option is to get a USB audio amp and then plug your headset into that, obviously a bit more expensive but it might allow you to get better sound out of the gaming headset.