I just had a read of the link, quite an involved process with plenty of option for error.
With care taken (note mistakes made/reported in link, try and learn from them)it should be okay though.
I think the best LEDs will probably be hi-brightness white, like this 5mm is probably okay depending on the width of the CCFL holder (the thing he drilled holes in) otherwise 3mm is the other common size for leaded LEDs. Or jusy buy a (white) LED headlamp and nick the LEDs out of it, may be quite cheap that way but LEDs may have shorter lifespan if poorly made - Nichia are a reputable manufacturer.
You need to ascertain what voltage/(waveform if PWM as Matt mentions) you will be using, and design accordingly. Depending on supply voltage, options are to parallel them, preferably with a series resistor in each leg to limit LED current, or series with one resistor, or series/parallel. Look like he did something like this in the link, they are in parallel and I see no resistor or mention of current control, probably not "ideal" (note comment about firing one LED into his chest on over voltage, which is a reasonable indicator there was no current control in place :-) ).
Or you could consider a constant current white LED driver (probably some ready rolled modules on eBay also), which is a better solution as it will ensure the LEDs are driven correctly. If you set it up right from the inverter supply/control, the dimming will still work. The driver chips usually have an option(s) for PWM dimming.
Make sure heat is dissipated adequately also.