A very long while ago I posted some solutions to Lights Out; back then, I solved the $n$-by-$n$ board by row-reducing an $n^2$-by-$n^2$ matrix.
Since then, both Boris Okun and Brent Werness pointed out to me that I should've solved Lights Out by using a scanning algorithm: propagating the button presses down one row at a time, and exponentiating the propagation matrix to make sure that I don't get stuck at the last row.
This is much faster.
With this method, here is a (scaled down, auto-leveled) 2000-by-2000 solution:
And here is a (very much scaled-down, auto-leveled) 5000-by-5000 solution: