Why does Miss Bingley invite Elizabeth to remain in Netherfield?

Jane Bennet originally goes to visit Netherfield at the invitation of Miss Bingley. Mrs. Bennet is excited at the invitation, seeing it as a great opportunity for Jane to get to know the dashing Mr. Bingley a little better. She urges Jane to travel to Netherfield on horseback; that way, if it rains, Jane will get to spend the night at Netherfield, giving her more time to spend in the company of Mr. Bingley.
Mrs. Bennet's devious plan comes to fruition, but not quite in the way she intended. Jane does indeed get to spend the night at Netherfield, but it's only because she got soaked to the skin on her way there, riding on horseback in the open air during a heavy shower of rain. What's more, she's caught a nasty fever and so has to stay in bed.
When word reaches the Bennets of Jane's illness, a concerned Lizzie immediately heads off on foot to Netherfield, tramping through the mud to get there. Though somewhat taken aback by Lizzie's disheveled appearance, Miss Bingley agrees to let her come in. Lizzie spends every moment she can at Jane's bedside. Jane is much comforted by the presence of her sister and asks if she can stay the night. Miss Bingley graciously accedes to her request and invites Lizzie to spend the night at Netherfield.