Have been having the same issue for the last few months on my laptop. However, I've noticed that connections with the IOS and Android apps are stable and do not suffer the same online/offline problems. Certainly not to the same extent as the PC.
I updated iPad with the latest version of RU that you released yesterday and it still doesn't work with ios 9. It gets further than it did in the previous version in that the app starts and you can log onto the remote PC. However, the screen on the remote PC seems to have frozen and any sort of action, such as CTRL-ALT-DEL, crashes out back to the main start screen.