Sorry but this isn't going to work.
There is no such thing as an HDMI-VGA cable, at least not a passive one. VGA is analog, HDMI is digital and they can't be directly interfaced without an active adapter. You would at least need to get an active (powered) HDMI-VGA adapter that is HDCP compliant, even this may not work properly with the Firestick and if it does the resulting image quality might not be that great.
That's a pretty old projector and it may just be time to move on to something newer that not only fully supports the current formats but has superior light output and resolution.