@fratscha Sorry for all the trouble!
I'm not currently aware of any issues that might block the iOS app from running after security checks. Has anything changed since the problem began? Able to test the iPad through a different network?
I am also having the same issue.
I have an iPad Mini 2 running IOS 10 and have been using the LogMeIn app for a couple years now.
Recently I had to re-enter my credentials (maybe after an app update?) and it asked me to verify with Captcha.
I click the Captcha check box and all it does is spin and spin and spin until it fails.
It never gives me the images to check.
The only way around it was to reset my password.
I can login to the web page using Safari on the iPad and the Captcha works there.
So something seems to be wrong with the app.