It depends on the kayak. I own a Hobie Pro Angler 14, Stealth ProFisha 525 and an Ocean Kayak Ultra 4.7.
With the Pro Angler, I use a Reliable Fishing Products kill bag. Keep multiple frozen water bottles and fish stays fresh all day.
ProFisha 525 has a fish storage hatch built in. Same thing - frozen water bottles.
Ultra 4.7 has a fish cooler (sold separately) - water bottles again.
If these don't work or the fish is too big, check out the tail leash sold by Red Legs Gear out of HI.