Assuming you mean the Libby by OverDrive app, it appears it requires iOS 10 at least, to run.
The iPad 2 from 2011, cannot update beyond iOS 9.3.5. It simply lacks the hardware, RAM and CPU to run anything newer.
The only option is to bite the bullet and upgrade to a newer model. I understand it still works, and there's no need, but after 12 years of function, it's not unreasonable to start thinking about moving on.
You can get an 8th generation model that can run the most current iOS version for $269 from Apple's refurbishes online store: Refurbished iPad - iPad - Apple