I'm also a big fan of Ashton's, HOWEVER you specifically inquired about a B & B "near the French Quarter or the West French Quarter"; West French Quarter not being a locally recognized term. Ashton's is 10-12 relatively short blocks Northwest ("Lakeside") of the French Quarter, a pretty significant hike.
You should know that in addition to hotels, New Orleans has inns, guest houses and bed and breakfasts. Because of relatively strict local regulations requiring commercial grade kitchens for the service of "full" breakfasts, many of these properties, particularly those in or very close to the French Quarter, do not have the "full" breakfast component; Ashton's does.
Although I don't have a lot of first hand knowledge regarding these properties, you might want to check out Soniat House and Audubon Cottages, though both lack the "full" breakfast component. Innkeeper's referal to the bbnola website is also very well placed. One B&B that might meet your location, bedding and breakfast criteria is La Maison Marigny, though I also lack first hand knowledge of this property.
Edited: 12:11 pm, January 30, 2013