Summarize:

Soldering involves melting metals and, naturally, you don't want to breathe them in. Open all the doors and windows around you. Get some air flowing through your workspace. A nice pair of polycarbonate glasses are essential when welding. Since you are working with small wires, you probably won't be able to wear gloves. Use tweezers to hold the wires and put the hot soldering iron in its stand when not in use. Never rest the hot iron against your work surface. Twist the parts counterclockwise to remove them. Set each part aside and in order so you know how to reassemble the pen. The bottom section of the pen is the battery. The part above it connects to the heating element, or coil. Notice how the wires run from the battery to the heating element. Again, make sure the pen feels cool before touching it. If a wire is broken, the pen should not heat up at all. You should see a red wire and a black wire. Bring the frayed ends of each wire together. Wrap the ends around each other to twist them into 1 wire again. The twisted spot needs to be thin so it fits in place. If you need more exposed wires, use wire strippers or a utility knife to cut off some of the casing. You can also rub the exposed filaments to fray them and make them easier to twist together. The red wire connects the battery to the heating element. The black wire is the ground wire and runs from the battery to the outside of the heating chamber. First, pick up a spool of solder wire from the home improvement store. Unspool it, laying the end of the wire near the wires you wish to solder. Turn the iron on and, after letting it heat for a minute, dab the tip against the end of the wire. Hold the same-colored wires together. With your other hand, bring the soldering iron over to the exposed portion. Dab the soldering iron against it to drop off the soldering material. Get more metal from the wire and continue applying it until the exposed wires are covered. Always move the soldering iron. Don't let it hang over the exposed wires, or else it might begin to melt them. Use the tip to spread the soldering material. The joint will begin cooling right away and solidify after a few minutes. The best way to cover the joint is with a shrink tube from a home improvement center. Wrap it around the joint, then apply a little heat to shrink it. A hair dryer is a good, safe way to do this, although you can also do it carefully with a match or the soldering iron. Another way to cover the joint is with electrical tape. Rearrange the pieces of the pen, packing the wires into their chambers. Turn on the pen and, if done correctly, it should work again. If it doesn't, return the pen to the store to have a professional look at it or buy a new one. The wires may have come loose from their connections. This is hard to fix at home, especially if you can't locate the terminals for the wires. Bring the pen in to a professional.
Work in a well-ventilated environment. Wear safety glasses to protect your eyes. Unscrew the vape pen parts. Twist same-colored wires together. Heat the soldering iron and dab it against a wire solder. Solder the colored wires together. Cover the soldered joint after it cools. Test the vape pen.