The wiring kit is just a complete package, convenient for some, but sounds like you may not need a kit since you understand the install and could piece the kit together, or even have parts in your "basement". The Kits come in different price ranges due to what is included, most come with a 20ft 8 gauge power wire, 3ft 8 gauge ground and a 15ft 18 gauge remote turn on wire. Others add a MANL (mini fuse holder) and the fuse itself, speaker wire and RCA cables.
As for changing the speaker wires, obviously if there is any damage to them, water, age, etc you would want to change, but if they are good then you can keep the stock wires in place. They will be fine with this set-up, or if you feel like changing it just because, you can.