It sounds like it could either be a cracked breaching piece (between the to taps where the spout screws on) or it could be the spindles leaking.
There is a o-ring in the spindles that stops water from coming out of the tap casing when the water is on. to check the o-ring take of the handles and cover flange turn the taps on with the sprayer turned off, is water coming out of the taps?
If not you can either cap off the fitting where the spout is connected and turn the water on and try and here/see water leaking or hook a pressure gauge up to the spout connection turn the water on then back off and see if the pressure drops when the water is off showing a leak in the breaching piece. How to pressure test
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
Your tub space is odd looking, It appears that your tub had an oval bottom.
For standard tubs The tubs are set in a base of foam or sometimes a cement type product to provide a solid bed for the bottom of the tub to rest on.
This is done to support the bottom of the tub and keep it from flexing.
If it was not there the bottom of the tub would be free floating and the flexing could, over time, cause issues at the mounting points on the rim of the tub thereby causing loose and or noisy connections.