It was the first bag my father bought my mother; I want to use it again!
Take a pair of scissors and cut the lining out of the bag
If it's vintage and has that kind of sentimental value, I would take it to a reputable tailor and explain what I wanted done. I'd be too nervous to try anything myself, even though I sew.
Tailor or a leather/shoe repair place. Also, a reputable dry cleaners might know who to have work on it. Make sure whom ever does work on it has a good reputation - it might be worth a lot!
Try to resew the lining back and it will be good as new!