Flying a kite is much easier if you have:
1. A good way to control taking the line in and out -- how about a fishing reel?
2. Leverage to guide the kite and something to absorb shocks -- how about a fishing rod?
I first saw this as a kid at Jones Beach. One day, there were a bunch of people "fishing" with their lines going up into the sky, rather than down into the water. It works great! It's even a nice alternative you can switch to if the wind is good and the fish aren't biting.
Step 1: Tackle Choices
The type of rod and reel doesn't seem to matter much, but here are a few guidelines:
1. Offshore reels work best for large kites, but both baitcasting and spinning reels work well.
2. Conventional monofilament line works fine; avoid tapered "fly fishing" line (no pun intended). Use the lightest line that you're sure can handle the force, usually 12-30 pound test.
3. Especially if you're using a spinning reel, connect the kite with a swivel to avoid line twisting.
4. The rod should be relatively stiff for the best control, but a little flexing acts as a nice shock absorber and makes flying kites in gusty winds easier on you and the kite.
In summary, size your tackle to match the forces that flying your kite will impose. If you're really just taking a kite along in case the fish aren't biting, size the kite to your tackle.