And turn it on again

But thank you for the suggestion!

Does it have a light showing HD activity?

is completely powerless.

After a few minutes, put the battery back in and see what happens.

And make sure it's plugged in to the power supply when you turn it on-- could be something as simple as a dead battery.

And also try unplugging all devices (mice, printers, etc.) and try again

plugged into it but the problem DID seem to start with the mouse.

get a screwdriver small enough to get at the battery.

asking for ID and password?

seconds before the logon screen, but now that's all I get.

That screen to which you are referring is part of the POST (Power-On Self-Test) process.
There's a guide here on troubleshooting this issue.

Normally, you would switch the machine on and start tapping the F8 key. A quick Google says that this should work for your machine but you may need to search the exact model if it doesn't.

If you can get it to load a black & white screen with boot options, select 'Safe Mode' and wait for it to churn through some screens. If this works, there is probably some conflict in programs or services trying to load at start up. Not necessarily a fix, but could give a clue on what to try next.