I wouldn't assume that the artist will demand money, or much of it. People give copyright permissions all the time (e.g., translation permission) without requiring a payment, and if the artist understands that this is a book of poetry and not a Coke commercial she may well agree for little more than a promise that you try to keep cobwebs off the original.
Apropos Bill's musing about the right to destroy an artist's work, perhaps you can hold the painting hostage and threaten the artist with a bonfire unless she agrees to let you use the cover?
But seriously, Bill, it is thankfully not always the case that the owner of a work of art is free to destroy it. Check out the
Visual Artists Rights Act.