You can buy 12V CCFL driver modules from ebay, but you probably need to get roughly the right wattage one.
But if you are reusing laptop screens its best to pick ones with a LED backlight. Easy to drive with just a DC voltage and they are much more efficient (Less heat, less power draw, more brightness)
Replacing this one with led backlit one doesn't help much in this case. In LED ones you have both power and data on one connector, this would require another logic board capable of doing this.
Replacing those two caps is not really hard, and does not involve any high voltage, as they are on low voltage side, and stupidly placed just above two rectifying diodes heatsink. So after some years electrolyte in these caps dried out. Just replace them with ones with the same capacity, and at least the same voltage. And make sure they are low ESR.
I found out some info on these lcd screens, and I think you might need to reconsider the whole thing. Your old lcd has 5V power supply, and 2 channel 8 bit interface, and your laptop screen is 3.3V, 2 channel 6 bit interface. You can find this information
here and
here. This site has a lot of useful info, good to keep it in bookmarks.