Agree with everything Pax said, including that if you're not willing to train, you shouldn't be using a firearm.
That said, the woman described does not sound to me like a candidate for a shotgun. After I bought a Mossberg 500 in the wake of a neighborhood murder when we lived in Kalifornia, I hired a police firearms trainer to work with both of us in the use of a 12-gauge. She shot it fine, but hated the recoil. A 20 would certainly be easier to shoot, but the issue of recoil remains, tho to a lesser extent.
My wife has been to the range with me a number of times, and can shoot all my handguns well. But again, recoil is her issue, and the only gun she really enjoys shooting is my Smith 22a. She is not a tiny woman and exercises daily, so strength is not an issue.
All that leading up to the idea that a .22mag revolver or semi-auto might provide our subject with a relatively effective round in a recoil-free weapon. Because of the lack of recoil to deal with in returning to target, it could be the easiest to learn to use effectively. I understand the feelings about sub-caliber guns for SD, but having and being able to use a .22mag would be better, I think, than trying to utilize a gun for which our subject appears to be ill-equipped, both by stature and inclination.
Having children in the house is an issue, but it's the same issue with a .22 or a .50; access must be denied to young children, whatever the gun you choose. Price is also in the ballpark, with Taurus and Charter (I think) offering wheelguns for .22mag. Smith has two, I believe, but both are higher in price, at least when purchasing new.
IMHO, of course.
"The gun has been called the great equalizer ... It insures that the people are the equal of their government whenever that government forgets that it is servant and not master of the governed." --Ronald Reagan