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.
Connect one end of the USB cable to your computer, but let the other end not connect to your iPhone. Turn your iPhone off.
When the phone is off, hold down the home button and connect the other end of USB cable (which should still be attached to your computer) to the dock connector.
Alternatively you can download recboot to fix this issue.