The Full Story
As Mussels says, there's no need to take the wheel out to repair a puncture. Just release the brake cable on rim brakes to give room alongside the brake pads and remove one side of the tyre (non-chain side with the rear) and you'll find you can take out the tube and feed it around within the frame/fork during a puncture repair operation. As a junior in the trade I had to do all the punctures that came in and only ever removed a wheel if replacing the tube, (and not always even then, but that's another story).
You'll need a puncture outfit, from any bike shop. With the bike upside down, take the non-chain side of the tyre off the rim, starting at the valve point, then pull the tube from the rim and free the valve from the rim so the tube sits between the wheel and frame.
Now pump some air into the tube and you'll hear it escaping from the spot where it's punctured. Clean the tube area at the puncture point with the little bit of sand paper that's in the puncture outfit. Then select a suitable size patch for it and take off any protective paper or foil from it's contact surface. Open the tube of rubber solution and put a little spot on the patch surface and quickly smooth it over the surface with a finger tip. Don't dally since it will start setting quickly. Then do the same on the punctured area of the tube, covering an area a bit bigger than the patch. Let the rubber solution completely dry on both.
When they are completely dry, centre the patch over the puncture hole and press the patch firmly in place, it will bond instantly. Take the small block of French chalk and either rub it against the rasp area on he puncture outfit box or scrape it with a knife edge to let a shower of a few chalk particles cover the patch and solution area. Smooth that over the area so there's no sticky surface left that might glue inside the tyre.
Pump just a little air into the tube to give it a tiny amount of pressure and ascertain there are no more leaks. The traditional way is by passing it through a bowl of water to let any stream of air bubbles show up. A less messy way is to use the lips, the most sensitive part of your skin, by passing the tube in front of your lips about half an inch away. The lips can detect the slightest air stream. Do this over an area covering a few inches either side of your original puncture since a sharp object can cause more than one puncture due to the tube shifting as it deflates.
Now release most of the air from the tube. Place your finger tips of one hand inside the tyre against the inside of the tread area and rotate the wheel right round, feeling for what caused the puncture, nail, stone, glass or thorn. If and when you locate it, prise it out from the tyre tread side.
Finally place the tube back into the tyre and put the tyre back onto the rim, starting opposite the valve point and working the tyre beads right into the well of the rim at that point. When the last bit is on at the valve point, pushing the valve in a little bit if necessary, make sure the tyre is sitting evenly all around the rim, then pump it a little, checking that the tube isn't trapped under a tyre beading. If all's well, complete the pumping.
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