Hello a week ago I was given to fix a Toshiba Satellite C55t-C5300 laptop, which happened the same thing, when you turn it on, it asked for a password. After a week of research with my cousin Alberto Concepción, we were able to find the solution.
You must first extract the BIOS from the board with a heat station, and through an interface read the firmware, done this using an exadecimal reader you can search the line 000080B0 shows you the password of your BIOS mine was "12345678".
Hope this can help you.
Vladismer Nápoles