I ran into something similar with my Lightning iPad and the Apple USB 3 Camera Adapter.
In my case, the firmware update didn't install immediately either. It just kept saying an update was available, but nothing happened for quite a while. What eventually worked was leaving the adapter connected, keeping the iPad on Wi-Fi, plugged into power, and letting it sit for a while. I also restarted the iPad once before trying again.
If it's still stuck after several hours, I'd suspect either a temporary issue with Apple's firmware delivery or a problem with that particular adapter. Trying it with another Lightning iPad or iPhone is a good idea, and if an Apple Store is nearby, the Genius Bar should be able to confirm whether the adapter updates normally.
Unfortunately, Apple doesn't publish firmware release notes for these adapters, so it's hard to know exactly what changed between 1.0.1 and 1.0.5. If you do find a solution, I'd be interested to hear what finally worked.