I haven't done it on the ST1300, but I did it on a friends Suzuki Bergman.
Some things to consider are how often do you see yourself removing the rear luggage? Do you ride without the rear trunk?
If it was me and because of how much I remove the rear trunk I'd keep the electrical connection outside of the bike, but would choose a connector that can stand up the elements (and some di-electric grease for good measure.) If you don't plan to remove it very often, you could hide the connection under the seat and not worry about the connector as much but it will be a pain to remove the top case because you need to remove the seat each time. I would wire it so that the wire came out the back by the rear plate (I haven't tried this location before so you should double check clearance, the right and left hand sides do have ample clearance as I've tried this and routed wires through here before.)
Its best to source a connector where the connectors are hidden or protected so that if the bike is turned on without the top box connected that you won't accidentally short the connector. This is also important if you plan to ride without the top box connected.
If you plan to ride without the top box connected, I would ensure that you can get a plug that can isolate the electrical contacts from the elements or source a plug where having power applied to the connector will not cause any additional corrosion as that can cause you to replace the connectors yearly depending on the environment in which you ride and how often you ride. Being in Florida you will have saltier air and corrosion will be more of a concern than if you were more inland.
Before starting I would consider how much power the various lights will add to the load of the load of the system. These days this is likely not a concern because LEDs can consume so little power (especially compared to bulbs) for how much light they put out, but something to be aware of.
The easiest way to connect up electrically is a T-Tap, it basically lets you splice into an existing wire. Some of the T-Taps require you to still connect to it via a spade lug, if you've never done this before look at the type and how to use it before you purchase it / install it.
You can make your own wires by putting all the wires you need into an electric drill and slowly turning the drill on, as the drill rotates slowly feed it more wire. The rotation of the drill will automatically twirl the wires and allow you to have nice all together wire for all of the individual wires that you need. You can also source wires that are already insulated together and have a thinner but wider profile to make it easier to fit under the seat.
I know I didn't really give you a how-to on getting it done, and hopefully someone here has done it before so that they can give you specific wires etc to you, but hopefully this helps you figure it out.