I do a lot of kayak fishing. I have seen many anglers hit the water with enough gear to stock a tackle shop. I too carried way more than necessary. It gets in the way, and it drags your boat down, making you work twice as hard to paddle. For me, there are two things I do to minimize my gear effectively....
1. Try different style/techniques from your kayak, and figure out which ones are most comfortable and effective from your yak. Certain techniques that I use from shore or boat do not quite translate to kayak fishing, for me personally.
2. Know your water! Now that you know how you can/can't fish from your kayak, bring what you know works on that particular body of water. For example, a lake that I fish often and know pretty well, I know that it is shallow, and that I catch most fish on wacky rigs, keel-weighted swimbaits, and the occasional chatter bait. Not to say they won't hit anything else, but based on my experience, those baits work best. I usually only carry 3 different colors of each. I know the depth, therefore I know what sizes/weights I'll want.
I carry Two plano boxes that conveniently fit behind my seat, and I can easily access them without tipping, or making any noise. I have a large ziplock bag in my milk crate full of bags of extra rubber baits, and a couple of back-up tools. THAT'S IT!!! I don't have any gear in my lap, or on anywhere on the deck in front of me when I'm fishing. I know what I need to bring. I leave a little bit of room in my boxes to bring a couple of experimental baits. Oh, and I should also add that I bring NO MORE than 3 rods with me. I have to make due with what I have. That goes along with rule number one, which i talked about previously. Hope this helps. Good luck!!!