If you use a dishwasher, I cant imagine why you would need any soap :S
That would do the trick for the board (just the pressure and hot water).
*word of caution- the larger and thiner the plastic the lower the tempurature in which it will warp*
For the keys just let them soak in a gallon ziplock bag or small bucket with a bit of soap. It takes only a few minutes. You can use hot water or cold (hot water (from the tap) works much quicker.
I use Comet for the keys (no longer for the boards!) a tiny bit in a bucket with keys will usually clean them in minutes. Then rinse them off and they will look like they just came out of the factory!!!
If you can do it by hand (and don't mind) using rubbing alcohol works insanely well! 70%, 90% whatever you happen to have. Just dab some on and rub it off and ....TAAADAAA! uber new!
experience-
I have cleaned 250+ boards and
tens of thousands of keys
tip you can practically clean anything (keyboard related) by just letting it sit in warm water for a couple of hours and agitating every once in a while.