Hello there, mmgvatx.
It sounds like you have a few issues happening at the same time with your iPhone 5. If you're not seeing your device in iTunes, the following Knowledge Base articles offer some great troubleshooting recommendations depending on the Operating System your computer is running:
iPhone, iPad, or iPod not recognized in iTunes for Windows - Apple Support
or
If you don't see your connected device in iTunes for Mac - Apple Support
Once you get your iPhone recognized, if it continues not to power up, then this information should help further:
If your iPhone, iPad, or iPod touch doesn't respond or doesn't turn on - Apple Support
Plug in your device and let it charge for at least 1 hour.

After a few minutes, you should see the charging screen.

If you don't see the charging screen within an hour, or you see the connect to power screen, check your connector, USB cable, and power adapter. Make sure that everything is plugged in firmly, free of debris, and not damaged. You might want to try a different USB cable or power adapter.
If your device still won't turn on, see what to do next.
Thanks for being a part of Apple Support Communities.
Kind Regards.