What plants are they infesting?

Insecticidal soap is a contact insecticide and has little or NO residual effect. So you have to spay all day long everyday!!!

Since concentrations of soap in Insecticidal soap over 1 percent are no more effective then the 1 percent solution, mixing anything stronger is a waste of your money. A solution much over that very effective 1 percent can cause problems with any plant the solution is sprayed on.
To control Whitefly requires more than just spraying something, and since the Whitefly has so many generations per year they have developed immunities to most all of the commercial insecticides the average person can get, and even many of those only licensed pesticide applicators can.
You need a totally integrated control program which includes the Insecticidal Soap, Yellow Sticky Traps (some people will try to tell you that these are for monitoring only, but each Adult Whitefly that is stuck on that trap is one that will not breed), and a good, in depth look at the soil the afected plants are growing in to see what the problem with that soil is.

What plants are they on?

And although the following may sound like I'm being exceedingly picky questions, I need to know more. Namely,
-- What do they look like.?
-- Where are they on the plants?
-- What does the damage look like?

I ask because once a week insecticidal soap should have done the job.

Once a week insecticidal soap does not do the job as I've been battling them all summer myself.

I tried the yellow sticky traps and ended up with a poor bird stuck in it. I had to slowly remove him from it and he lost quite a few feathers. He appeared to be okay, but it was really heart breaking to know that I put something out there that put him in danger.

I guess I'll have to loo into soil issues now. Does anyone know what types of soil issues can encourage them to multiply?

Soil that cannot support healthy plant growth may predispose some plants to whitefly infestations. If the plants are constantly in some kind of stress mode or are over fertilized (which is a kind of stress), whitefly and other pests....which might not ordinarily be too interested, might consider moving in and taking over. So, what Kimmsr means, is that by taking good care of your soil, you'll be doing a better job at taking care of your plants.

However, there's a long list of plants that are PRONE to whitefly, are known whitefly hosts. If your garden is stocked with several of those plants, you can simply expect those pests, no matter what kind of condition your soil is in.

Whitefly have natural enemies called Encarsia formosa, they will eat the larvae of the whitefly, these encarsia will greatly decrease the whitefly population, and not pose a problem for you. You can purchase them online, I just don't know how. It is just a suggestion, we had to use beneficial insects in the greenhouse I used to work at, and the encarsia were helpful creatures. This link may be helpful.

Here is a link that might be useful: Encarsia

I think that importing beneficial insects works much better in an enclosed environment rather than in the great outdoors. It's a much better idea to make sure your garden areas are simply a good environment for all kinds of critters, and hopefully, plenty of beneficials will move in. That's sure how it works at my place!

Rhizo has made an important point above. White fly needs a host plant(s) to reproduce. Your job is to figure out what you are growing that is the major host plant. Do a search for "white fly + host plants" for some help in plant identification. Remove the problem plant and that should end your white fly problem. If you live in the south variegated privet is a major host plant. California has many types of host plants as you will note by searching that state.

To date my yard had been free of white fly until early this summer when I discovered the gardenias covered with white fly. I did a search to see which of the newly introduced plants might be the host culprit. Found it right away...arugala. Ripped them out, bagged them tightly and tossed into the garbage. End of whitefly problem.

I have, over the years, tried many methods to control white fly. The only successful one has been to fill my hose end sprayer container with lemon detergent which contains no additives such as degreasers and antiseptics. Setting the dial at one tablespoon per gallon I spray the affected plants early every morning and this controls them for a 24 hour period. Nothing else has worked for me.

What did you use on that sticky trap that caused a bird to become adhered? This information from the University of California - Davis may be of some help understanding the problem and what to do. As is stated in the article White Fly are resistant to most all pesticides and those pesticides will kill off the beneficial insects that could help control them.
The reference to your soil means take a good look to be sure you are growing your plants in a soil, that for them, is a good, healthy soil and will provide the plants with the nutrients they need, when they need them, and will not over feed them.

Here is a link that might be useful: White Fly control

"As is stated in the article White Fly are resistant to most all pesticides"

Funny how the above statement, from the UC-Davis site, actually says "Whiteflies are not well controlled with any available insecticides" and also says "If you choose to use insecticides, insecticidal soaps or oils such as neem oil may reduce but not eliminate populations."

The link has a lot of information on white flies, it is worth reading...just dont trust someone elses interpretation of what it says, read it for yourself.

Schmoo