There is a fuse in the wire between the start button/switch and the starter solenoid. With no sound as you described, either the fuse holder disintegrated, one or both ends are corroded or the fuse blew. Bring a new fuse holder down with you next time, find it and replace it.
How long to run? You need to know about your energy budget (what comes out) and how large your alternator is, and then do the math.
Energy budget forms are available here: Energy Budget
West Marine used to have one on their DIY page, but I just checked and it disappeared! Dratz, they used to be good...