The sprint button on the Xbox controller is activated by pushing down on the left joystick. This is a mechanical switch that is depressed, and will usually make a click sound when activated. The button can have a mechanical failure, where the button is not being depressed, or an electrical failure, where the switch is not making contact. The fix described here is related to an electrical failure of the switch, but also briefly describes the mechanical action of the switch.

Tools Required
Torx-8 security screwdriver. This fix requires you to open the Xbox controller. This requires a Torx-8 (T8) security screwdriver. If a security screwdriver is not available, a regular T8 can be used, but the security pin inside each screw will need to be snapped off.

Continuity Tester. We need to check and see if the switch is conducting when depressed. You can use an ohm-meter for this.

Soldering iron and de-soldering tool. This fix requires removal of components from the circuit board of the Xbox controller. This requires a soldering iron and a de-solder pump, or de-solder braid. If you are not familiar with de-soldering equipment, it may be best to ask someone with experience to remove the components from the circuit board

Step 1: Open the Xbox Controller

Remove the 7 screws from the bottom of the Xbox controller. 6 of these screws are visible, 1 is hidden under the battery pack, under the serial number sticker.
If you only have a T8 screwdriver, break off the security pin inside each screw. I do this with a hole punch tool. Pushing the tool into the screw head bends the center pin. Once the pin is bent, push the tool into the screw head beside the bent pin, and it snaps off.
Once the screws are removed, take off the plastic cover of the controller. Remove the circuit board. There are 2 vibration motors attached to the circuit board. Remove the connectors from the circuit board so that the motors are not attached to the circuit board.

Step 2: Check the Mechanical Action of the Switch

Locate the left joystick mechanism. To the side of the joystick mechanism is a small switch underneath a plastic post that sticks out from the joystick.
Depress the joystick and watch to see that the switch is being depressed. If the switch is not being depressed, then there is a mechanical problem with the joystick. You may need to check the joystick for broken pieces.
If the switch is being depressed, then we can continue to check for electrical conductivity.

Step 3: Test Electrical Conductivity

Turn the circuit board over and locate the pins for the switch. See the picture below. The pins are circled in red. Conductivity is measured between the pins marked by the blue lines.

Connect the continuity tester to the pins marked by the blue lines and depress the joystick. If there is no continuity, then there is an electrical problem with the switch. We need to remove the switch and replace or repair it.

Step 4: Remove Joystick and Switch From Circuit Board

To de-solder the joystick and switch from the board, we need to gain access to the pins. They are currently convered by the plastic housing for the trigger switch. So, we need to remove the plastic trigger switch first. The plastic trigger is held in place with 2 plastic clips, and a 3-pin potentiometer, shown in the red circle.

Turn the circuit board over and locate the 3 pins for the potentiometer, shown in the second image.
De-solder these 3 pins, then remove the entire plastic trigger from the circuit board.

Turn over the circuit board and locate the pins for the joystick and switch. They are 4 post pins, 2 sets of 3 pins for potentiometers, and 4 pins for the switch, as shown circled in red in the third image.

Step 5: Remove Joystick Base

Now you have the joystick and switch removed from the circuit board, as shown in the image.

Turn the joystick over and look at the bottom plastic base. Locate the 4 bent pins holding the base in place. Bend these pins out straight so that the base can be removed.

Once the pins are straight, the base can be pulled off the joystick. The switch will stay in the base.

Step 6: Replace or Repair Switch

Here is the switch. You can remove it from the base and replace with a new one, if you can find one. Or you can repair the switch.

To repair it, you need to take it apart. This causes the switch to fall apart. Once it is apart, it does not snap back together.
You can use the joystick housing to hold the switch together. The housing has 2 pins that hold the switch in place. These can be bent to press down on the switch so that it stays together (shown in the second image).

Step 7: Replacing the Switch

If you want to repair the switch, do this:

Use a sharp knife to cut off the 4 black plastic rivet heads on the top of the switch.
Take the square plate and switch button off. There is a small round metal plate inside the switch. Remove it.
On the bottom side of the round metal plate, it is probably corroded. Scrape it clean with a small knife.
Put the round plate back into the switch, then the switch button, then the square plate.
Test continuity of the switch. If it is working, put it all back together.

Step 8: Reassmble the Joystick

Place the switch back in the joystick plastic base.
Place the joystick back in the plastic base and bend the 4 housing pins back into place to hold the plastic base.
Bend the joystick housing pins that are located over the switch, so that they hold the switch in place, and hold the switch together.
Place the joystick and switch housing back onto the circuit board and solder all 14 pins.
Place the plastic trigger housing onto the other side of the circuit board and solder the 3 pins of the potentiometer.
Put the circuit board back into the controller, and re-connect the motors to the circuit board.
Close the controller and re-insert the screws and tighten. Make sure the controller housing has a good fit. This is important so that all the buttons make good contact with the circuit board.