Step 1: Make a Jig
I started with a piece of cardboard (real cardboard, i.e. single layer) from a piece of packaging. Do not use corrugated paperboard of the sort used to make shipping boxes.
I cut mine into 3 pieces after seeing that the thickness of the cardboard (0.44mm) was about 1/3 the height of the LED (1.5mm) that I was trying to solder. The LED here is a 1206 part, but this will work for smaller parts as well.
Each piece will become one layer of the jig, with a hole in each of the top two layers to hold the LED.
First, cut a hole in two of the pieces with a sharp hobby knife. The hole should be just big enough to hold the LED with very little slop. The bottom piece doesn't have a hole, and prevents the LED from falling through.
Then laminate the pieces together using a glue stick, lining up the holes in the top two pieces. Press until the glue stick has set with your fingers or a heavy book.
Step 2: Prepare the Magnet Wire
It's easier to do either of these jobs when the wire is in a loop and taped down to the cardboard.
So measure out a piece of wire that is about 2cm longer than twice the length you're trying to make (assuming that you want both wires to exit in the same direction). Bend the wire at its center into a loop whose diameter is about the LED width, and tape down the non-loop part so that the loop will extend just past the LED when it is taped down as well.
Then strip (scrape or burn off the insulation) and tin (apply solder so it coats the bare part of the wire) the wire where it will be attached to the LED. If you use a knife to strip the wire, try to find a dull blade and drag it along the wire at a shallow angle so you don't nick the wire itself (which can lead to wire breakage). If you're using sandpaper, wrap it around a matchstick or fold it so that you only have a small amount of sandpaper contacting the wire (in other words, don't strip any more than you have to and don't damage the wire).
Though this picture shows the LED in place, if you're scraping you probably want to wait to insert the LED until after you use the sandpaper or knife to prepare the wire.
In the first picture, I've used my soldering iron to strip and tin the wire after I taped it down in a loop.
If you're soldering these into a series string, just tape the wire down so it is across the LED electrodes, rather than making a loop in the wire.
Step 3: Insert the LED and Solder the Wire to It
Then tape down the loop or other end of the wire so that the part you have stripped and tinned is against the LED pads. You can tighten the wire by pulling slightly on the free end of the wire. Make sure that there is no (or at least very little) space between the wire and the LED.
Now solder the two together using a fine-tip iron and a very small amount of rosin-core solder. You may want to use a bit of rosin flux on the pads. Don't use acid-core solder or flux!
Step 4: Trim the Wire and Remove the LED
Or if you have a pair of good cutters you can use them, and reduce the chance of cutting into the jig. But since the jig is just a piece of cardboard, if you tear up the jig you can make a new hole...
If you've used a loop, the excess part is the loop itself; if you've soldered across the bottom of the LED for a series string, that part is the part between the two LED pads.
Press straight down with a sharp knife until you go through the wire. Then remove the tape.
Step 5: Twist the Wire (if You Want) and Mark Polarity
If you bothered to place the LED in a particular orientation and so know the polarity, you can mark the +/anode wire by following the wire from the known end of the LED.
Alternatively, you can use your ohmmeter to figure out which lead is which after you twist the wires together.
If you want, you can twist the wires together. The easiest way to do this is to hold each wire between thumb and forefinger of each hand and twist both wires in the same direction at the same time.
If you have a single LED wired to two wires (instead of a series string), if you put your ohmmeter on the "diode" range (the arrow with a bar symbol) and connect to the two wires from the LED to the two meter probes or clips, you should see an open circuit indication in one direction (on my meter, it reads 0L), and something else in the other direction.
Diode test ranges usually show the forward voltage drop across the diode at a low current. So in this case, I see that my LED has a forward voltage of 1.836 volts at the test current (this is a very old and inefficient LED; typical red or green LEDs should read around 1.4V to 1.7V). When you see a non-overload indication on your meter, the read/plus lead will be connected to the positive/anode lead of the LED. Mark one of the LEDs (in this case, I just bent the anode lead of the LED; in other cases I've used a black sharpie marker to mark the negative/cathode lead of the LED).
You can often see the LED glowing as well when you get it hooked up the right way, as in this picture.