Unable to log in to Find My Friends

I changed my AppleID password and so now "Find My Friends" requires me to log in again. It remembers my username and I enter my password (which works to let me log into e.g. appleid.apple.com or into iCloud from the System Preferences on the same iPhone). I am then told that as I have 2 factor authentication enabled that "Verification is Required". I then get a screen giving me the choice between verification via SMS or with my iPhone. Regardless of which I choose I am stuck on this screen. "Next" does nothing (it changes colour when I tap but otherwise nothing). If I choose "unable to access device..." I can type in my recovery key but "Next" does nothing from that screen either.

I have tried resetting my settings but it made no difference.

I have tried leaving it 24 hours but the problem remains.

I am using an iPhone 4 running