Can anybody think of why my iPad has 13 app updates that don't install? Whether I touch Update All or one at a time, nothing happens.

This is coming up quite frequently these days.
Check the following things first.

Are you downloading on wifi or 3G? If its on 3G check the file size as there is a 50mb limit I think.
Check your free space on the iPad settings>general>usage.

Can you download the updates individually? In the app store or do they all not work.

There is many different fixes for this so you can try a reset etc, but I just deleted the app that would not update and downloaded it again via the app store under 'purchased'

This should resolve it, but beware as it may delete your saved files so just try it on an app that's not important.

When I did this to just one app they all updated again without a hitch.
Hope this helps.