My iPhone 5 has a broken power button and now won't sync through any cable or to any USB port (I mention this because helpful would-be troubleshooters keep telling me to try different cables and ports, like I wouldn't do that to begin with). When it stops charging I'll be screwed, but I hope to nudge it through until the next generation comes out and I'm eligible for an upgrade, because I am currently monetarily deficient. Fortunately, with a little imagination, so far I HAVE OVERCOME.
Broken iPhone buttons
Jailbreakers have a lot more options here. The rest of us can dink around with built-in accessibility features when our buttons break. See how to do it here. This solution leaves you with a floating ghostly gray button that gets in the way, but I've adapted to it much like you do when you add a passcode to your phone originally. This solution also still lets me take screenshots and such.
Syncing music without cables
I think I'm one of the few humans left who actually used a cable anyway to sync, but I have a complex system of iTunes playlists that depend on play counts and last played dates and skip numbers and probably the phase of the moon. I've had them so long and implemented so many kludgy solutions that I'm afraid to look at what I've set up.
Wifi sync is an obvious solution, but I've never been able to get it to work. In fact, I don't even know anyone who has.
I enabled iTunes Match and found it sucks way less than it used to, so that's solved. I actually haven't found any issues with it yet. The play counts are automagically updated, as are my playlists, so that's all good.
I made a Photo Stream stream called "Save" to which I send iOS photos I want to end up in iPhoto. They appear there when I start iPhoto. I import them. This is a good system because I never, I mean never, hook my iPad up to my laptop, so now my photos are actually more organized since I can send photos from any device to that stream.
Side note: I discovered that you can turn off automatic import to Photo Stream in iPhoto preferences, and this for me really cleans up this "sync" process. I litter my camera roll and, therefore, my Photo Stream, with screenshots and snapshots to remind me to do things, and I don't want to sort through that junk to find the photos I want to keep around. Moving them from my iOS device to a separate stream and disabling the rest works great.