As everyone else said, there is no scientifically sound way to effectively reflect a sufficient amount of light from Mercury to Saturn.
But you're in luck since Russian scientists already found a solution to your problem: Project Znamya (meaning "Banner) they constructed a 65-foot-wide sheet of mylar that could be unfurled from a central mechanism and launched from the Mir space station.
They basically sent a giant mirror in orbit around Earth, in your case it would be orbiting around Saturn or its satellites.
The experiment did in fact work but to have a system that could actually be used to lit up the whole planet they still had a long way ahead
First of all the size of the mirror
The plan was to first test a 65-foot mirror (Znamya 2), then a 82-foot
version (Znamya 2.5), finalize the test phase with a 230-foot mirror
(Znamya 3), and, eventually launch a permanent 656-foot space mirror
installation that would be capable of fully turning early night in
Russian cities into something close to full-blown day.
And then the scale of the system
"The scheme called for a chain of many satellites to be placed in
sun-synchronized orbits at an altitude of 1700 kilometers, each one
equipped with fold-out parabolic reflectors of paper-thin material,"
Crary writes. "Once fully extended to 200 meters in diameter, each
mirror satellite would have the capacity to illuminate a
ten-square-mile area on earth with a brightness nearly 100 times
greater than moonlight."
And we're talking about Earth. Saturn has diameter 9.5 times bigger than our planet and a surface 83 times bigger. So of course your satellites will need to be way more that what you'd use to lit up Earth and probably bigger as well.
For a more in depth article: The Man Who Turned Night Into Day