I have difficulty connecting smaller plastic rocker switches. With these switches the metal contacts can be pretty small, so it is not like you can attach a big heat sink in there. When I go to solder wires onto the switch, if I get the whole system (connector and wire) hot enough to melt solder, then the switch melts. In one case I was soldering and the contact just came right out of the switch because the plastic that was holding it in melted.
1 Apply a small amount of solder to the clean soldering iron tip. This aids thermal transfer. You will see this mentioned in all guides on "how to solder" (if you don't see it mentioned, find another guide).
2 Tin the wire you are going to connect. Go "ouch! ouch!" and remember to use something to hold the wire (a small vice is handy) so that you don't burn your fingers next time.
3 Apply a small amount of solder to the clean (yes, you may need to clean it again: get used to doing that frequently) soldering iron tip.
4 Place the wire on the connector on the switch.
5 Apply the soldering iron so that it touches both at the same time.
6 Wait one (1) second at most and then feed the solder into the gap between the soldering iron and the parts to be connected. If the solder does not melt straight away, remove the iron and peer at where you tried to apply it - maybe the angle wasn't quite right.
7 Remove the solder. Remove the soldering iron.
8 Inspect the joint when it has cooled down (say, 20 seconds for larger items). It should be shiny and with not-too-much, not-too-little solder. There must be no pointy bits hanging off it or shorts to other parts.
Cleaning For cleaning the soldering iron tip, I have found that the curly brass sponge stuff is better than a wet sponge: the latter cools the tip down too much all of a sudden and tends to leave dross on the tip.
Temperature If the solder on the tip of the iron looks "off" (becomes coloured or matt) really soon after applying it, then your soldering iron is too hot. If solder does melt as soon as you apply it to the tip, the soldering iron is not hot enough.
Solder Solder with lead-containing solder with flux in it (e.g. multicore 60/40) is easier to use than lead-free solder. Do not use solder or flux intended for plumbing - they will destroy things.
Cleanliness A good solder joint needs the surfaces to be soldered to be clean. You can get flux pens which dispense a small amount of flux when you apply them to the surfaces to be soldered. This is in addition the the flux which is in the solder. Flux chemically cleans the surfaces. It is generally a nuisance to remove afterwards, but you really should.
Fumes The flux will give off fumes. You don't want to breathe them in or get them in your eyes as you will get a sore throat and eyes. Arrange for some sort of ventilation that keeps the fumes away from you.
It should be possible to use a brush-on flux to improve wetting get fast reflow to the surface. By heating the wire and feeding solder to it only, it should flow to the terminal quickly. This may require a wider tip to maintain a more constant temperature or a regulated heater tip iron.
I would expect with flux pre-applied to both surfaces and resin-core solder the solder time should be < 5 sec and preferably 3sec with an adequate power tip.
I known that 5m LED steel leads have a heat velocity of about 1mm/s and thus 3 second latency with 3 sec. solder times at 3mm below the base of the LED will just reach the epoxy. If you hold it any longer the low-temp clear epoxy will ooze around the leads and damage can occur.
Using this experience I can appreciate your challenge.
Without doing further research I would suggest with the proper tools and technique it should perform well in less than 5 seconds.
Another method Pre-solder-tin the wire around the shank of the insulation and the edge of the terminal. Then insert, twist and squeeze. This also improves the stranded wire rigidity with solder in the shank. By the time the solder melts it makes contact with a small gap to the terminal, reflows with wire solder feed, then you remove it quickly.