The most important part for fixing the scratchy keys is cleaning all the dust/dirt out of them, not just adding lubricant. To accomplish that you need to disassemble all the switches, and then figure out how to remove all the grit from the top housings. Just a detergent soak doesn’t get it all. The easiest effective method would probably be to use an ultrasonic cleaner, of the type used for jewelry or lab instruments (note though, I haven’t tried this). But those are pretty expensive. You could probably get an okay result by carefully swabbing each one out with a q-tip or similar. After you clean all the grit out, then you might want to add lubricant afterward.
To be honest, the switches in an AT101W aren’t worth that kind of effort. You should just desolder them all and swap in brand new Matias switches.