I don't know what you mean by "the little charging home button" or what reaction you'd expect from that. So I'm going to ignore that, but it's possible that if I knew what you meant it would change everything I write below.
The simplest explanation for this is that the screen is broken. If your computer responds when you plug it in then it's still powered-up and the mainboard is still live. If it still makes notification sounds that could confirm it, so try calling it and see whether it rings.
Not having a working power button makes it hard to do anythihg else. If not for that I'd suggest pressing and holding the power button to force a restart: if you didn't see the splash screen that would confirm that it was a hardware problem (while if you did see the splash screen and then it went black it would mean it was software). But as it is I can't think of a way of forcing a restart unless you have USB debugging enabled already (letting it power down then charging it is a little risky: if it recognises that the power button is broken it might restart automatically, but if that doesn't work you are stuck).
My best guess is a hardware problem, in which case repair is the only option. If your computer can talk to it you might be able to back up some stuff off it (e.g. photos), but if it's set to connect in "charge only" mode I don't know how you can change that without being able to see the screen. I have recovered data from a phone with a broken screen, but that involved using a wired connection to mirror it to a TV (and I don't know whether your phone supports such a thing, as I don't know what phone it is) and a lot of trial and error (display screen, decide next move, swap screen cable for a keyboard, tap cursor keys, switch back to screen to check results, rinse and repeat). If you were able to do something like that to change your default connection mode to file transfer (which may require going into developer options) I could see a way of backing-up data to a computer, or installing and launching something like WiFi File Explorer or AirDroid, but setting anything up will still require interacting with the phone with the screen not working, so won't be easy. So it may also be that your only option is to get the phone repaired and ask them not to reset the phone in the process if there are data you wish to recover (which an independent repairer may do, official repair services tend to do a reset anyway as part of their process and may ignore any request not to).