I concur with Frank and Glenn regarding the use of tethers in current. I would discourage you from doing so.
Any line long enough to wrap around your wrist or ankle might do so in the confusion of an unplanned wet exit in moving water. If the line is under considerable tension, as it might easily be if your boat hangs up and you are in the current, you might not be able to free yourself even if there is a quick release device on one end. You could be in a situation in which you couldn't reach the line to cut it, assuming you had a knife, if you are suspended in strong current.
I have taken many more swims in whitewater than I care to admit, including more than a few in which I was paddling alone. There have only been a couple of times where I had to let go of my boat, or accidentally did so, usually when a cold or wet hand slipped off.
I have never used painters on kayaks, but do have them on all my river canoes. There is no reason you couldn't have one attached to your kayak, but I would be certain that it is securely stored under a shock cord as Glenn suggests. In my canoes I usually loop the painter under a pair of shock cords atop my deck plate. It should be easily accessible from the end of your kayak when upside down. This might require you to drill some holes in your boat to mount shock cords.
I find that for the purposes of river self rescue, a relatively short painter about a boat length long suffices. With practice, you can learn to just free up a short length of rope to give you something more convenient than a grab loop or toggle to hold on to while you swim your boat to shore, or you can free up the whole length if necessary.
At times it is not that easy to swim a boat out of strong current, especially if loaded. It is often much easier to swim yourself into a nearby eddy, then reel the boat into it using the painter.
1 Canoe/Kayak Trailer
Kayak & Canoe Outriggers
Kayak Deck Gear Bags
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