You have a fair bit of room inside that plug, you can't really go wrong unless you let stuff overheat. see if you can get a little solder on the top of the plug joint, then a little on the wire - then hold them together and apply the iron for 3-4 seconds. Assuming the iron is hot enough, that should be enough to melt them together. It's intimidating at first, just take your time. it only takes a few seconds to get things hot enough to melt the solder, so don't leave the iron on anything longer than about 5 seconds - once the solder starts to run, remove the heat and it will fuse as it cools. try to stay away from the plastic housing, obviously - but it wont matter if you accidentally touch the iron to the plastic - the only way you can screw this up is to get the pin too hot so that it melts the plastic housing - you would have to leave the iron on it for longer than 5 seconds for that to happen. look at the other joint. You're trying to get it to look like that. It looks doable.