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.