There are different standards to pipes and threads. Your issue is that you are trying to mix two different threaded standards.
The ball valve that you linked to has national pipe threads (NPT). NPT are tapered and it is that taper that makes the seal between a male and female joint. No washers are required.
What is shown in your photo is compression fittings which have a straight or running thread. The seal is made by either a ferrule or a rubber washer.
In your case, the connection between the copper tube and the supply hose has been made as follows:
1/4" Copper Tube > 1/4" Compression by 1/4" Compression Coupling > 1/4" Supply Hose
The seal on the copper tube is made by a ferrule that is located under the nut. As the nut is tightened, the ferrule is compressed between the coupling and the pipe.
The seal on the supply hose side is made by a rubber washer which is usually built-in the the nut of the supply hose. (The compression coupling would have come with two nuts, one of which has been discarded as the supply hose has one built-in.)
So, to answer your question, you simply need a 1/4" compression x 1/4" compression valve that will replace the current coupling. Here is a link to such a valve.
I'd use a small tubing cutter to cut tidy/clean ends on both ends of the 1/4" copper pipe ...
... cutting off whatever portions of the 1/4" copper pipe are kinked or deformed. I'd then slip on a compression fitting called a 'union' ...
... which requires tightening with a pair of wrenches. Wikipedia has a general article on compression fittings here. You-tube hosts a concise 3 minute how-to video regarding installing compression fittings on plastic or copper pipe. The copper tubing portion of the demo starts at 1:45 ...
http://www.youtube.com/watch?v=eQGL8MBLlaE
Small tubing cutters such as the one pictured above will cut diameters 1/4", 1/2" and 3/4" and anything in between. The small size of that cutter design makes it excellent for tight confines. It would be the first one I'd buy for my tool box. You could also use a medium size cutter, such as the one used in this 90 second tutorial ...
http://www.youtube.com/watch?v=nLB3MomrXd0
The small cutter works in the same way. There are two main points to keep in mind when using either of these tubing cutters ...
1) Tighten the cutter only a little bit each time, making a number of light scores around the pipe instead of several deep ones.
2) The cutter wheel needs to continually track in the previous score, so keep a close eye on the score and make sure the score is not spiraling along the length of the 1/4" copper pipe, which can happen if the pipe is bent or deformed in the area where the cutter is being used.
After the cuts are completed, the cut ends of the 1/4" pipe need to be very close to circular (for example, not deformed into an oval). If they are not circular, then even if the ferrules can be forced onto the 1/4" copper pipe, the fitting will leak. If the ferrules do not slip on easily, it might be due to the ends of the 1/4" pipe being out-of-round, or (unlikely) due to defective ferrules. Using a tubing cutter will help ensure the cut ends of the 1/4" copper pipe are circular.
Alternatively, a hacksaw (or similar) can be used to cut clean ends on the 1/4" pipe. This will likely result in ends with sharp edges and a burr. Remove these with sandpaper. Clean ends will allow the ferrules slip on without gauging, which would lead to a leak. A fine-toothed file can be used instead of sandpaper, but the copper dust/shavings will most likely clog the teeth of the file.
Here is a 7 minute how-to video from England regarding leaks at compression fittings:
http://www.youtube.com/watch?v=aez4ctMtbuI
Best Answer
I would minimize the amount you bend them. Think of it like a metal coat hanger, if you bend it in the same place several times, it will break. Only the copper pipe is hollow, so it will break a lot sooner. As best you can, avoid bending it multiple times in the same place.