Hi, I am A. Stevens here to help with your question!
I think the last option for you to do is to try a Factory reset of the phone - if in fact you can get the menu to come up.
Press and hold VOLUME UP and Home(center) buttonPress Power until the phone vibrateWait until you see the Android logo then release all the buttons.Select Wipe data/Factory reset with VOLUME DOWN, press Power (right hand-side)Select YES -- delete all user data with VOLUME DOWN and then press PowerAfter format, press Power again to reboot phone. (select reboot system now)
If you are unable to access this reset menu, I would have to advise that your phones hardware has gone defective.
I have just had a look at your suggestion. I also just bought a new battery to see if this would make a difference. However, we seem to be stuck, when turning the phone on, with just an image of a grey battery on the screen with a black background and a white circle in the middle which presumably is supposed to indicate something is happening. But this doesn't change.
So given this is the case and we can't get to a menu, do we basically just have to replace the phone?