The most frequent short in a cord happens near the point where the cord enters the mouse. Usually cutting 6 inches of cord, stripping it and resoddering will fix the mouse.
If the mouse uses a specific connector, it's easier to leave a few inches of wire attached to the connector and reconnect to those wires instead of trying to redo the connector.
Wire strippers (You can get by with a lighter, but it's a lot harder).
Shrink tubing, or electrical tape
Volt Meter (optional but helpful).
I'm sure there's a few youtube videos you can find, but the process would be:
Cut a chunk of the wire out from the mouse. Inside the mouse there should be a point where the wire splits and leaves the outer shell. This is generally where you want to cut the wire. By leaving the smaller leads, it gives us a section of wire to work with when resoddering. You'll want to take about 4-6 inches of wire out of the cord.
If you have 2 wires of the same color (usually black), at the connector, take exceptional care to track the 2 separate wires. This can be done with a volt meter, or physically following the wire. The actual wire should have 4 individually wrapped wires, and an outer unshielded connection. The other unshielded connection is the ground and will likely be the same color as one of the inner wires where it meets the connector.. You don't want the unshielded wire to be able to touch your circuit board and ground out the board. If your working with a large section of exposed wire, it's worth slipping a section of shrink tube around it.. Grounding out the board won't damage anything, but certain aspects of the mouse may be unresponsive as a result.
To make sure your connecting the correct black to black, you can use a volt meter.. Before cutting the cord, disconnect the wire from the circuit board by pulling the connector off. Touch the meter to the outer connection of the USB cord, and then to the various sections of wire exposed on the connector. By running electricity through the wire the meter will complete the circuit and tell you when your touching the same wire. Make note of which part of the connector it's hooked to. You can do this test to the wire after you cut it by simply connecting to the end of the wire instead of the connector.
After cutting the chunk of wire out, strip each wire and make a small hook at the end. Do this for both sides of the wires your reconnecting. A physical connection makes it harder to break the sodder joint later on.
Slide a section of shrink tubing over one of the wires, then link the 2 hooks together and clamp them down. Apply a light sodder coat.. Less is better when using sodder. It's not strong enough to warrant applying a huge blob. Slide the shrink tubing over the new connection and heat the tube until it shrinks. This can be done with a heat gun or a lighter. Repeat this process for all 5 wire connections.