Well, the first thing to do, is to dry out the floor surface and more importantly the subfloor, so that the boards don't continue to warp. In fact if you have a plywood subfloor, it going to be very difficult drying the underside of all the hardwood floor boards affected, because of the waterproof nature of plywood.
It's very important to dry the floor quickly, not just to stop the warping, but also to stop the mold that my get a grip on the wood. Mold spores can be left even after the floor dries, and may cause health problems for some sensitive individuals. Note how I say : "May cause" and "Some individuals".
The easiest way to dry out the top of the floor is to buy or rent large fans and point them down to the floor. Keep the AC on if you have it, but open windows next to the floor area about 2 inches. The excess moisture has to go somewhere, even with air conditioning. Now more importantly, below the floor in the basement (you didn't say it you have a crawl space or full basement), you also need to have large industrial fans pointed up to where the water has come through. If you happen to have a finished ceiling below the floor, you might also consider cutting a neat (and repairable) square in the drywall. This is so the fans can better dry the under side of the subfloor. It would help if a few holes were drilled up into the subfloor where the leak began, to make sure a puddle of water is not sitting between the floors.
You should also ventilate the basement a bit, unless the weather is really humid, and then use a dehumidifier in the basement on those really damp days.
You can expect the wood to continue to swell, and if it gets to a point where you simply cannot walk across it, you should then relieve the stress by removing about 2 boards all the way across the middle of the kitchen. It's a good idea to do this anyway because this will allow more ventilation of the subfloor. I assume you will call your insurance agent, but you might also call in a wood floor contractor. He will have an electronic moisture meter. You can test some of the good wood in a dry part of the house to determine it's EMC (equilibrium moisture content) and then measure the EMC where the flooding has occurred. Keep records of these figures with the dates. The hardwood AND the subfloor have to come back to the normal EMC before you allow anyone to resand the floor.
Some insurance companies will insist on removing all the damaged floor right away. This might be a good thing, if they cover the whole cost. They might be aware of possible mold problems, and are willing to set you straight right away. Or they may want to wait to see what damage remains permanent. Keep you receipts for fan rentals as this should be covered also. Up north we experience less problems with mold because of our long heating season, which really dries out the wood.
Even if the insurance company wants to remove all the damaged floor you will still have to monitor the subfloor until the EMC gets back to normal. Only time and ventilation will do this. It may take as much as a month or more for this to resolve itself.
If you find that the damage is minimal, because the water drained out well, and you have dried the floor quickly, you still should wait about 4-6 months before sanding the hardwood floor flat again. If you sand the floor before it's had a chance to fully dry, the cupping (concave) you see now, will straighten itself out after the sanding, and create a crowned (convex) floor instead. Then you will be forced to sand the whole floor yet again. So don't be rushed by the insurance company or a flooring contractor into sanding the floor too soon. Understand that sanding a severely cupped floor will remove as much as a third of the total life of the floor. There is just so much wear layer on the top of the groove. You might ask the insurance company to compensate you for this also if the damage has gone into another room, and that room has to be fully sanded and finished.
So in conclusion, dry the floor quickly above and below. Check and recheck EMC of wood and subfloor. Remove a few boards to relieve stress. Start repairs after heating season has begun and you are positive it all dried out. Watch for signs of black mold forming. If insurance is covering the cost of the damage, keep good records. The job will consist of a combination of repair and replacement. Read my free article in this site about how to hire a flooring contractor. You can also read the free repair article, to get an idea of how this task is done. The insurance companies don't always provide the best of floor guys.