Hi,
I made a note some decades

ago:
It needs to be fully charged, then push reset and the start slider at the same time for a few seconds.
Does the reader has a SD or Memory Stick Duo Card inside? Remove them and try again.
I always charge with the sony charger, not via USB. The charger has a yellow plug (they are color coded?). Outer minus, inner plus, 5V, 800mA, 7W.
If you connect it to the computer you should find at least a folder "database" and "Digital Editions". When it is empty, I think the firmware is missing.
When you google "prs+" which is an alternative software for PRS-Readers, you find a google Code Archive. On the fourth page of the "download"-section is an "uninstaller_505.exe" for uninstalling prs+ and reinstall the standard sony software. Maybe this can be helpfull to restore the original software. But the battery must be charged!
If you are able to restore it - it is a wonderful reader

. With PRS+ even better.
Good luck!
EDIT:
Nowadays it is complicated to get a working battery because of a missing thermistor (I know for sure about that for PRS350 and 650, not sure if the 505 has that problem too). But it is possible to change the battery on the 505. There are Youtube videos about that. I've done it one time but it is not easy and one have to be very very careful when removing the case because of a thin and delicate data cable.