Wire strippers, get ones for the wires you are working with. Most household repairs and appliance repairs are probably for something small.
Shrink tubing
Solder
Some wire
Get as much access to the wires you are fixing as you can. You will need to get your hands and some tools on the wires. So loosen up the harnesses or unmount them from how they are attached, etc.
Plug in the soldering iron. The cheap one from Harbor Freight will work, but be careful about it pulling itself off of a table.
If the wires are too short, you will need to have a scab or extension to get to the right area.
So in a setup with a scab, you now have 1 in place wire that is cut, and 1 extension scab. 4 ends to be prepped.
Cut off about an 1/8th of an inch of the wire jacket on each of the 4 ends.
Twist the braided wires inside so you don't have any wild ends sticking out.
On each end hold it to the soldering iron when it is hot for 10 seconds or longer until it will accept solder. Sandwich the wires in the middle between the soldering iron and the end of the solder.
If you don't have helping hands, you can use the soldering iron stand to hold the iron. Then hold the wire in one hand and hold the solder in the other. Don't have your fingers too close to the wire ends... it's hot!
After the solder has penetrated the braided wire, it is now tinned! Don't put too much on. If there is too much, get it hot again and quickly flick it off.
It is very probable that the wire jacket receeds a little bit while tinning. This is expected and okay.
Cut some heat shrink at least 1/2 an inch that will cover the soldered area and some exposed wire plus another 1/8th of an inch on each side. Slide it on at least 1 inch past the area to be soldered.
To be continued...