repair - Part of the power connector broken and stuck inside laptop power jack - how to fix?

I have a Samsung ultrabook (the Apple-clone type that is not easily customizable and looks like it was made out of one part) and the insulating (wider part, not the actual center pin wire that carries the electricity) part of the power connector stuck inside laptop power jack.

I can still charge the ultrabook by holding the center pin wire at a certain angle, but it requires holding and applying pressure in a very particular way.

I tried using a very fine pipette to remove the residual insulator part from jack, but it's stuck and the friction of the pipette and the stuck part is too little.

I don't think I will attempt to do anything further unless there are any recommendations here.

My main question is: Should I ship the ultrabook to Samsung (probably expensive) or will a cheaper supermarket or PC repair shop (potentially one online that accepts and specializes in laptops) likely be able to fix it?

Replace the jack where you connect the charger at once, just go to some laptop shop and they will open the screws and remove the broken part inside,

You an go to regular shop, the problem is not that big to go to vendor,

To remove a broken power connector from the power jack isn't hard to do. Simply, take some tweezers and/or couple slotted precision screwdrivers (usually found in a local hardware store) and carefully extract the remnant.

A temporary fix to the power connector itself would be in two delicate steps:

1. carefully insert the extracted remnant on to the power connector, taking special attention to prevent bending, twisting, or worse, breaking the copper lead inside the remnant, in the process.
2. Next, the pieced back power connector will need to soldered together to hold the broken power connector in one piece. Take an ordinary solder iron, some solder wire, and flux (also found in your local hardware store) and solder both pieces together, also taking care to prevent making a mess (since melted solder is hot that it could melt the plastic parts of the power connector if you don't put in your best effort to do the job, light and easy.). When soldering you always have to remember that less is more. For the start just put small, flat beads on the seams and add some more to fill in the gaps. Once the solder has cooled the power connector can be used as normal, but sensibly, treating it like an other delicate component. The crude weld will hold for a while before it will eventually break off. For an insurance measure, try a few practice solders with small metal scraps to be more confident on your final solder on the power connector. (Extra: try different combinations of different types of solder wire and flux to see which one will have a stronger hold.)

I have some experience with fixing delicate parts.

The idea of soldering the pieces might seem quite radical, but if you're busy or don't want to spend some extra bucks on professional repair, this suggestion could be one of the only things you could do in about an hour. Unfortunately, I don't have images for a visual, but sorting out the process into steps would make it a bit easier. Hope I have answered your question.

As always, test it out to see if the suggestion is viable.

Good Luck and enjoy some weekend time trying some stuff out! :)