My old iPhone 4s had a reboot loop and also the power button is broken. I tried to switch to recovery mode and that's how it worked for me after hours and month:
Let your phone run out of battery & download RecBoot
Update iTunes, start iTunes.
Press Home button while you plug in the USB to your iPhone/iPad
Start RecBoot and press "Enter recovery mode".
There will be several popups of iTunes asking for restore and so on. I installed a new blank iOS version, and it's working again, after 6 months!
Also there will be a new screen with "iTunes" and Lightning adapter—that's the right way.