You can screw down the upper parts of the keyboard, with the keys, joystick.
And you can wash with lukewarm water soap or other not agressive cleaning things can be used. (Without break the warranty label)
If many stuff between the keys, you can easy remove all keys from their places, but be carefully: do not break the fixing bows!
The joystick is can be easy disassembled.
After all keys removed, you can wash anything.
When clean, lets dry it, and reassemble. Keyboard layout photo good idea for find the place of all keys

Good idea to use silicone spray (can be bought in car shops for a plastic care), blow the downside of the keyboard top, to make lubrication for the keys. After this the keys can moving a very fine.

Some photos, how can you disassemble the keyboard:

Remove these screws:
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Remove the uperside of the keyboard from the machine:
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Machine without the keyboard, you can see the keyboard membrane below the rubber part:
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The keyboard:
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Parts of the internal joystick:
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Downside of the keys, with the fixing bows:
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Carefully remove the keys, and see what a dirt below the keys
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The removed keys:
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They are superb photos!

Wow! Free puzzle game for Enterprise owners! Pity that cannot be emulated by PC emus.
As for Easter, cleaning the Enterprise: puzzle and sprinkle! No rabbits are allowed near the EP because it'll need recleaning. (And don't boil Easter eggs on the heatsink.)

But seriously: Happy Easter!