tomscharbach Dell's recommendation (echoed by the boards) is to try a hard reset:
Thanks for your welcome advice. Unfortunately, this problem seems to indicate that it's caused by one of the other zoo animals.
I couldn't lift the battery out of the laptop, for some reason, so I disconnected the battery cable from the main board. Then I followed the rest of the procedure, reconnected the battery, replaced the bottom cover, and tried again. The results were the same -- 5 seconds of keyboard LEDs followed by 5 seconds of fan noise, and then nothing at all.
These refurbished Dells are usually so reliable that I'm willing to buy her another one. This is the first E5570 we've gotten, and it's clearly a reduced price version. The "E" in its product number must stand for "economical." Lower resolution screen, i5 processor, more plastic. It worked well for her games, while it lasted, but I just ordered another 5580 like mine. The difference in cost is probably well worth paying.