Hi,
Are you using the tip the same as the picture?
If so you really need a tip that looks like this, a chisel head.
The tip in the photo is about 6mm in size.
I've re-soldering lots of speedos and have never had a difficulty, using a tip similar to the above.
What method are you using?
Try this. Get the iron up to temperature, 350 - 400C will be more than enough. Melt some solder onto the tip. Then clean off the tip on a wet sponge. Allow the iron a minute or two to come back to temperature. Put fresh solder on the tip. (You should get a small puff of smoke as the flux burns.) Then de-solder the wire from the speedo. Repeat the steps for each wire. This is all assuming that you have the correct size of tip on the iron.
Before replacing the wire. Remove as much solder as possible from the PCB (printed circuit board). (But be careful not to over heat the board). Use solder sucker or a solder braid / wick to remove the old solder.
Remove some insulation from the new wire. Twist the wire tight. If required trim the wire, so that it fits the hole in the PCB. Hold wire in place on PCB, resolder, feed in enough solder to that it flows up into the wire as well as enough to hold the wire on the PCB.
If anything above is not clear just ask.
MiCk B. :-)