[Hamoth]

We can ignore her barking for about ten minutes before it literally becomes too physically painful to ignore. We tried that for a week without any difference at all.

I hate to say this, but you are going to have to endure. 10 minutes is not at all sufficient for the ignoring treatment. You gotta be prepared to deal with barking for HOURS on end for about month to get this behavior to go away. It will get worse around week 2, then week 3 shows a dramatic improvment (if your pup is like mine in that regard).

The reason that silence is being so hard to learn is that you have quite consistently rewarded barking behavior with attention and concern of various kinds including some negative things that will add a lot of anxiety to the mix. But on the other hand your ignoring is very inconsistent and short-lived. You have indeed taught your dog to keep barking to eventually get your attention. Your pup will need to unlearn all of that, and that's what makes this hard. 1 month of barking or a lifetime, that's the choice you are facing in my opinion.

In my opinion, A barking collar will not be effective in your case. It will just make things worse. I suggest a good pair of ear-plugs...a lot of patience, and the determination to stick with 1 treatment that you can chose based one evaluation of your dog and the methods there are out there.

The barking we tend to have a problem with seems to be more about what might be going on outside. She barks running from window to window or from the vacuum to the fan or whatever else in the room seems to bother her.

Like she's defending her territory from it? Even if it's imaginary threats?

She does know speak. Oddly enough, we can never get a full clear bark out of her when she speaks for a treat. The best we get is a slightly muffled woof.

Try saying "Speak", then knocking on the front door. Watch with wonder as the barking is unleashed. After about a week, you won't need to knock to get the barking going.

After half a week it still hasn't gotten to the point where she has a clue what the work quiet means.

Well, you sound like a reasonable person...you know the dog is smart...but she isn't learning "quiet" you might want to up the value of your reward, add the clicker-training (if you have not already)...Clicker lets you easily tell the dog exactly WHAT behavior is being rewarded. It bookmarks the thing that's getting the treat. For example...

"chewy, Speak!"
"bark, bark bark abrk...long pause....bark bark bark!" CLICK - Treat.
Good boy!
"Bark bark bark!"
"Chewy, hush!" (happy voice...)
"murfle...woooorrrrrr..bark bark bark..."
waiting...Eventually,
long silence...CLICK - Treat.
"Good hush!"

Repeat 30 times a day. 10 in the morning, 10 noon, 10 night, or more. I find doing longer sessions with Chewy helps him learn faster. For shake, the first day he got it, I had him do it 60 times in a row, treating him no matter how sloppy it was. The next day he knew it still and continues to do well with it.

The more valuable the treat, the more effective hte behavior and the more willing the dog will be to learn. Steak pieces but into small chunks the sizes of raisens are GREAT training tools for thos hard to grasp behaviors.

She is not at all an alpha dog. One of the first things she does once she is comfortable with a stranger (person or dog) is roll over onto her back and spread her limbs.

Mmm. She might not be alpha with strangers, but have you evaluated if she is alpha to you?