If it's molded-in & there isn't enough to access to solder, you could try opening the case, but price a replacement first, as you might ruin it trying to open it.
Expensive components like that are probably best left to someone who has some knowledge of opening the cases and making a repair, and then closing it up. I have done a similar repair and used silicone and a hot glue gun to make it weather-resistant once-again, upon reassembly.
I also side w/using a small soldering iron instead of a soldering gun, though there are small and large wattage versions of each, and that is really what you need to go-by. Something like this:
http://www.homedepot.com/p/Weller-40-Watt-Soldering-Iron-Kit-SP40LK/100391613#
And here is a 25 watt version:
For larger gauge wires, a gun like this:
100/140 watts two heat range gun. I have these and use them according to the task. This job would call for the 25 or 40 watt gun. You also might want to consider using a heat sink, if there's enough room to use one, to keep from feeding excess heat into the circuit board.
And, yes, no acid-core solder, rosin-core solder only.