Q: How can I delete the iBooks app on my iPad and re-download it?

You cannot. The iBooks app is a native app and is part of the iOS. The only way you would be able to put the app on the device would be to restore the device as new. I would see about redownloading the books.

You may need to contact iTunes support for help with the purchased books.  http://www.apple.com/emea/support/itunes/contact.html

Are you sure you're looking at your own Library and not the iBooks Store view?

The book summary sheet appears only when you tap on a book within the Store. If you tap on a book within your Library, the book either opens, or starts downloading from the cloud.

For what's it worth, when I upgraded to iOS 8.3, most of my books disappeared from my Library and I had to sync them back thru iTunes, or re-download them from my Purchases section of iBooks Store.

Bottom menu bar, My Books - those are your books stored locally on the device

SSorry, that's the bottom menu bar of the iBooks app.

it should be "my books", "featured", "NY times", ...

What collection do you have selected in My Books ? Tap the collection name at the top middle of the bookshelf to open a popup list of your collections, and select 'All' if that isn't currently selected