At the University of Texas in Austin, Professor Janine Barchas has created (or rather recreated) the art exhibit at the British Institution in Pall Mall that Jane Austen wrote about visiting on May 24, 1813.
What Jane Saw is a wonderful site and a wonderful resource for art exhibits during our period. Each room of the exhibit is faithfully (as far as possible) reproduced. The floor plan allows you to see the paintings as they were exhibited and each painting is clickable, taking you to a larger image and historical information. Moreover, it includes a scan of the exhibit catalog
This exhibit does not include the portrait of Mrs. Bingley that Jane Austen mentions finding in this letter, That portrait has been identified as Portrait of Mrs. Q (Mrs. Harriet Quentin) by William Blake. This was, obviously, not in the Joshua Reynolds exhibit reproduced on What Jane Saw. According to the letter, she saw this at at Spring Gardens.
But don’t let the fact that this exhibit doesn’t include Mrs. Bingley stop you from visiting. I dare say you’ll want to linger a while.