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If you don't want to replace the cable or install a new jack, you have a few of options:

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  Glue it into a coupler or a short extension like one of the following (with the coupler you will need another short run of cable on the other side):

  
  
  

  

  
  


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  Repair it with a zip tie. Use this guide for instructions: http://www.instructables.com/id/Repair-a-Broken-Ethernet-Plug/

  
  
  

  

  
  

It's entirely possible to cut off the plug and install a new one, but you will need a crimping tool. It's about $15. The thing is, your local store will probably charge you more than $15 for a cable - even if you buy it online, it'll probably be close to that when you take shipping into account. Therefore, it's cost-effective in my opinion, even if you only need it occasionally (and those pieces tend to break somewhat frequently, in my experience).

Also, if you ever need to run large amounts of cable, it's much cheaper to buy it in bulk and cut it yourself.

EDIT: Don't forget to match the plug type to the wire type - stranded or solid (in your case, the wire is almost certainly stranded).

For a quick fix, I've successfully used a small piece of paper to wrap the 3 non-conductor sides of the socket and then insert the cable into the socket. Leave some paper extending outside the socket so you can prevent the plug from just pushing the paper into the socket. This hack can work for years, if there is not a lot of movement of the device or cable.

Otherwise, the only true fix is to cut-off and replace the plug.

Or, replace the cable if it is not a difficult routing issue. Cables are pretty cheap if you shop around online.

You can always cut the broken end and add a crimpless plug. Here is an example. They are more expensive than a crimped end, but perfect for an emergency. Different brands/models have different capabilities such as self cutting the wires.

CHEAPEST AND EASIEST FIX. Slip a RUBBER BAND all the way under the plug clip if the locking plasticclip is cracked but not broken off. The rubber band will lift the inner non-broken part of the clip just enough for you to hear or feel the click when it locks in place. You can remove the plug by pressing down on the clip and rubber band as usual.

This works for a SEMI-PERMANENT FIX, in case you rarely re-insert the plug, to avoid having the cable plug annoyingly disengage when you accidently move the cable or router, especially when you are unaware of it.

I was looking for the same thing, you might as well benefit from my research. There's a whole bunch of third party products that solve this exact problem. In no particular order:

This is called an RJCLIP, you can buy them on Ebay and Amazon.
Heres a video demonstrating the idea.

This is an INTELLINET 771436 Repair Clip , Amazon has them

This one is a DELOCK RJ45 repair kit , Ebay has them

This one is called ClipFix , Amazon has them

Ive also found some asian ones but I cant locate them at present.

I 'fixed' my broken one for my laptop by attaching a long elastic band which I then stretched over and under the keyboard (luckily not blocking any keys) which pulled the plug in tight.

Then I splashed out on a short extension (about $5) as is shown in the article, and joined the two parts securely again by an elastic band, not by glueing.