This page says clay soil absorbs water at the rate of 1/2 inch per hour, 4 times slower than sandy soil or loam.
http://www.ornamentaledibles.com/tips/soil.htm
Let's assume you get 5-10 gallons per minute or 300-600 gallons per hour. A gallon is 231 cubic inches or 462 square inches 1/2 inch high. So 600 gallons/hour needs 277200 square inches or 1925 square feet. Assuming your trench is 12 inches wide, you need 1925 linear feet of pipe under steady state conditions.
Let's consider another situation where you store all the rain water and let it absorb as slowly as it wants. Assume you get 600 gallons which requires 80 cubic feet. So if your trench were 12" by 12" you would need 80 linear feet of pipe.
Obviously, the soil needs to be graded away from the house (in the back yard). If that means you need a retaining wall, then you need a retaining wall.
Regarding the swale, improved drainage sounds helpful. The real question is, why isn't water shedding (what is the real issue)? I've had swale issues, and the problem in those cases, was not with my swale, but the neighbors'... so water backed up in my yard. In such a circumstance, you can talk to the neighbors about the issue, ask them to fix it, offer to help, and/or sue them for damages.
But if the neighbors aren't to be blamed, then another alternative to keep the grass cut very short (grass slows drainage). Or you could check into a trench drain (concrete ditch). This is just another option that I didn't see mentioned, although it may not be your best option. As Tester101 pointed out, being there to see the issue is probably necessary.
Regarding power outages, gasoline or propane generators are the most reliable answer. A moderate generator should be enough for a sump pump, the fridge, and should keep the toilet flushing too. As you know, the drawback with a battery bank is that once it discharges, then you don't have a simple solution, like getting more gas.
Edit- Here's a possible way to use a retaining wall to increase the grade away from the house.
Best Answer
These sort of leaks can be notoriously hard to track, some of the possible causes can be
I noticed you said you have checked the meter for a whole house leak. That is good and I would recommend you do that by turning off all taps toilet cistern taps and washing machine taps etc, then go to your meter write down the reading and leave the test over an hour if possible as very small leaks can be difficult to detect on a whole house meter.