Sorry, I can't in all seriousness use your username, but here's what I've found so far.
As nearly as I can tell you'll need to remove the motherboard then turn it over and take off the audio circuit board before you can work on unsoldering the headphone jack. Here's a guide that should get you that far.
Dell XPS 15 (9560) Audio Circuit Board Installation & Video Repair Guide
This is going to require moderate soldering skills, so if you're not comfortable with that you may need to consider finding a shop to do the repair. Basically, on each of those 7 pins you identified you need to use a solder vacuum or solder wick to remove as much solder as possible from each pin individually until all of the legs are free within their respective through hole. At that point the jack should just drop out and you can pop the replacement in.
Speaking of which, I think I found one that should work, but since I can't see the other side of the jack I can't be 100% sure. This one is used on other Dell models, which is a common thing for manufacturers to do, so I'm hopeful about this one.
New DELL 5000 5555 5575 5758 3558 Audio Jack Headphone Microphone Connector Port
With any luck that'll get you going. You can take a chance and order the parts or go ahead and pull the motherboard in order to verify the part. If you do that, add a picture and we'll see if we can tell if I've found the right part.
Well rats, looking again at your picture, it appears the jack I found isn't going to work. On the side with only three pins, yours has one pin, then a missing space then two more pins, where the one I came up with has the missing pin first then three more in a row. I'll keep looking and see if I can find anything else.
A picture of the opposite side would still help though.
Here are a couple more that may or may not work; I can't be sure without seeing the opposite side and the front.
5pcs/lot Audio Jack Socket Connector for Asus Dell Lenovo IdeaPad 110-14ISK 110-15ISK etc Laptop Headphone Combo port 7pin - Price history & Review
You'll want to look at the one marked as part "Q" on this listing.
ChengHaoRan 1pcs Laptop Audio Jack Connectors Port Socket Set For Acer Asus Lenovo HP Dell Toshiba Samsung MSI
Okay, those pictures definitely help, but the bad part is that all they ended up doing was confirming that the ones I found aren't correct, darn it.
Do you have a pair of calipers? Any chance you can take measurements of the black plastic part of the jack, L x W x H? I found one that looks like a possiblity outside of the color (it's blue), and it might help if I can compare the measurements of yours with this one.
Laptop Audio Jack headphone Microphone 7 pin 7pin 7 pin 3.5mm Connector SMT SMD Socket for HP DELL LENOVO ACER ASUS 13x10x4mm
According to the listing, that one measures 13x10x4mm.
124 
44 
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1 Comment
@fartsaucethe3rd I can tey to help you find a replacement headphone jack, but I will need to know the dimensions of the jack, the amount of pins in the jack, and the amount of space between the pins of the jack, as well as a picture. Could you provide that info? Adding images to an existing question
by Andrew S