Unable to Install Update - iOS 9.3.5

I am having problems updating my iPad Air 32Gb

It is running iOS 9.3.3 and would not update to iOS 9.3.4. Now it won't update to 9.3.5 either.

When I tried to update to 9.3.4, I received messages saying that the verification of the update had failed. For 9.3.5, it just says "An error occurred installing iOS 9.3.5" but doesn't specify what.

For for both updates, I deleted the downloaded update files and redownloaded them multiple times, to no avail. It won't update via iTunes either. The only advice I can see anywhere about these problems are to 1) download the update again - done and failed. 2) refresh or restart the device - done and failed. 3) update via iTunes - won't work either.

I have several GB of free space and the problem occurs even when fully charged and whether plugged in or not.

My iPhone updated fine, apart from some glitches (with contacts and app crashing) that seemed to resolve themselves. I understand these updates are important security ones, so am keen to get it sorted out.

I would appreciate any ideas. Other than suggestions to do what I already have done!

Thanks.