What did it?
"turning-out the dining room" done it. It was the cause or reason for the women's fatigue or consequent condition. Done instead of did because the thing happened past tense. (If she were to use the word "did" even if she were aware that "did" was proper grammatically, her friend would think she was being uppity.)
After reading through the comments, I realized I had added "that" automatically to understand the sentence. I realized "that" I had added "that" automatically to understand the sentence. It is colloquial and uneducated but accurate. The first thing to lose in dialects is extra words and syllables. It's a lazy way of speaking.
I've also heard, ...it's turning out the dining room "what" done it. Using the same sentence construction.
For turning-out, think how did it turn-out? It turned out grand, a splendid event, everything refined, to put things in order. During the Victorian period turning out a dining room would have involved hand polishing real silver and brass. Perhaps they would launder the linens, tableclothes, placements, napkins, and lace dollies. China would be washed and replaced, the table might be set for diner. Curtains and rugs may be taken outside and beaten. Mirrors and furniture would be polished. If it could be done in a day even with the help of maids, it would be a huge undertaking. You could also turn-out the bedding which leads to the imagery of a pillow case being turned inside out and removed from a pillow.
In my mind, these women are speaking over the fence with laundry billowing, gossiping about the mistress of the house. It also is not the true cause but only an excuse to explain the women's condition. Being ignorant, Mrs. Sutton isn't privy to the real cause and can only create suitable explanations from her experience.