I bet it's that close switch

It's possible you could just soak it from the outside and hope it seeps into the switch underneath...but that's probably not the
best way to do it. DeOxit is made by CAIG Laboratories. The one you'll want is the D5.
That unit is a good sounding one, and I'm excited to see you get it up and running! The CD portion has a really nice sound, and using the CD-Out connected to a bigger stereo reveals how good the CD section actually is. Weirdly enough, the CD-Out also doubles as an RCA-in, if you ever wanted to play another device through it.
Once you get in there, take care with all the cabling. In reassembly, the most difficult and nerve-wracking part for me has been when cables are in the way, when closing it all up. I learned quickly to take LOTS of photos of how the cables are routed and how they're organized. One last thing...In my first repair, I broke a connector, having never seen one like it and not really knowing how it worked. I didn't realize most of them are spring loaded and release the wires, when pushed down toward the board.
Otherwise, you've got this! Keep us posted!