After removing a laser diode from a DVD Writer, instead of trying to solder the wires to the tiny cut off pins, try this.
Get a piece of stripboard/veroboard or even a piece of old circuit board with 2 parallel tracks.
Stripboard only needs to be 2 tracks wide by 4 holes long.
Fit the diode in to the item it's going to be used in. May be a heatsink or laser module with focus lens etc.
Support the stripboard and add solder to the tracks, both ends at the same time is a good idea.
Carefully bend the laser diode's pins apart slightly to fit over the tracks. As long as you are careful the pins are quite sturdy and can take a bit of pressure.
Then hold the laser housing/heatsink and solder the diode leads to the track side. No need to try and get the pins through the holes. They are fine on the track side. Then solder the laser power lead to the other end.
The lead I use is the CD Drive to motherboard lead from an old PC. They usually have 2 leads plus a shield around them.