updated iphone 4s, won't turn on?

I have an iPhone 4S, 16G, and today tried to update it to iOS 8.4.1 (I think that's the latest one?). I made sure I backed up all photos, etc before updating because I've heard that the update can wipe some data. The update appeared to work until I tried to turn my phone back on; no matter what combination of buttons (home button, lock button) I hold, my phone will display the Apple Logo for over a minute before going black again. I've tried plugging it into my computer, which only turns the logo on and off again - there is no sign of my phone being connected on iTunes or iPhoto however. I know it's probably just an old phone that's given up, but I'd really like it to at least last another couple of months until I can get a new one! Does any have any advice on how to solve this problem? Any similar stories? All help would be really appreciated 🙂