Here are your parts
Replacement parts for WHIRLPOOL DU840CWDQ3 | AppliancePartsPros.com
Will a new housing repair the leak or do I need to order the pump motor and assembly?
Only you can answer that question.
You have to figure out where the leak is coming from.
If the pump housing is cracked then replacing it would repair the problem.
I would also replace the motor seals. I would get Item 9 in section 5 as it looks like it includes both seals and the impeller.
If it is just leaking from the motor to the housing then odds are a seal kit would fix this.
It is often difficult to find the actual leak location because the water can run down the parts till it drips off so often the actual leak is higher up.
A small mirror and a flashlight can help locate a leak. An ideal mirror is the type used by mechanics witch has a small mirror mounted on an extendable handle. You could always just tape a mirror on the ned of a stick if you have to.
Note: It is a good idea to add a quart or two of water to the unit before firing it up. Dishwasher pumps do not like to run totally dry.