To refurbish this delicate bonnet, I very carefully detached the lining and the silk ribbon from it, and washed it again in a mesh bag on the delicate cycle. The lining was so fragile it tore apart as I was detaching it. The ribbon held up slightly better, but was showing significant wear around the stitching holes.
I have a significant store of scrap fabric, so I cut a piece of cream colored satin batiste, approximately 13 by 5 inches long. I folded it in half so it measured 6 1/2 by 5 inches, and made a French seam on one edge, then narrow hemmed the others. I gathered the seamed edge so it would fit into the bonnet, then carefully hand stitched the lining to the bonnet, gathering the lining slightly to fit.
Here is the lining stitched to the bonnet, and stuffed with batting to make it sit up nicely for the photo.
I still can't get over how well this cleaned up! I have some lovely 3/4 inch cream silk satin ribbon, now I just need to decide how to attach it. Do I want just some ties with maybe a rosette at each bottom side? Or should I gather the ribbon slightly to frame the face opening, as in the original? I have some cute little pearls I could add...