That sounds like a ‘retina’ issue, with the screen resolution on the iPad being much higher than usual. there are many imperfect solutions to that, but since you are using background-size, a nice solution is to save the image at twice the actual dimensions needed. To avoid a larger file, you can afford to save the image at a lower quality to compensate. That gets very nice results.
As for designing for the iPad, remember that this isn’t a very practical exercise. There are hundreds (if not thousands) of devices being used out there, and it’s totally impractical to try to chase them all down. Most people in the world are not using iPads. Despite all the talk about them, they are a niche item.