Just wrote you a massive explanation then closed my window by accident. Lets try again!
From what you say, with water running even when powered off, this is certainly a mechanical failure of the water inlet valve. There will only be a couple of possible causes for this:
1) The valve is being jammed open by an obstruction, possibly grit in the water supply (did you have plumbing work done recently or work on the water supply in the street? This would be a classic cause). If an obstruction is the case then repeatedly starting a fill cycle, draining and satrting the fill again MAY shift the blockage.
2) There is a failure of the valve its self. Essentially it's just a valve with rubber seals operated by a small electro magnet and sometimes the seals fail (often as a result of a poor water supply allowing grit into the rubbers).
If the information at (1) above doesn't help, you may have to replace the valve (Though in examining it when you come to take it out, you may be able to locate and remove an obstruction).
To help you out there is a parts manual for your machine available in PDF here:http://www.greatplainsapplianceparts.com/order/Parts%20Manuals/Asko/ASKO-Dishwasher/D3112%20(DW20.1)%20SS.pdf
Looks like the diagram on page 2 shows that you need part 135, although it seems to me that the author of the manual has got part 135 (Part No. 8073774 Cord Set) mixed up with part 140 ( Part No. 8073825 Inl. Valve Single 120V Sens.).
I imagine that if you go to the ASKO website and use the contact us option:http://www.askousa.com/customer-care/contact-us/
They should be able to clear that up for you
Hope this all helps